Arithmetic of Natural Numbers

Introduction Addition Properties of Addition Numerals Multiplication Arithmetic of Natural Numbers Bernd Schröder Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) 2. So we need arithmetic. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) 2. So we need arithmetic. 3. But arithmetic must be constructed within set theory so that it will be part of our framework for mathematics. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) 2. So we need arithmetic. 3. But arithmetic must be constructed within set theory so that it will be part of our framework for mathematics. 4. Moreover, it turns out that the abstract properties of the operations we consider will have far reaching consequences. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) 2. So we need arithmetic. 3. But arithmetic must be constructed within set theory so that it will be part of our framework for mathematics. 4. Moreover, it turns out that the abstract properties of the operations we consider will have far reaching consequences. 5. So we also need to prove some properties for the operations Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication The Peano Axioms Can’t Be “It” 1. The natural numbers are about counting objects, about adding and multiplying, subtracting and dividing, comparing numbers to each other, etc. (And the Peano Axioms don’t directly provide for these things.) 2. So we need arithmetic. 3. But arithmetic must be constructed within set theory so that it will be part of our framework for mathematics. 4. Moreover, it turns out that the abstract properties of the operations we consider will have far reaching consequences. 5. So we also need to prove some properties for the operations and in another presentation we will consider the consequences of these properties. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. A (binary) operation on a set S is a function ◦ : S × S → S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. A (binary) operation on a set S is a function ◦ : S × S → S. Binary operations do exactly what addition and subtraction do: They take two objects and produce a new one. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. A (binary) operation on a set S is a function ◦ : S × S → S. Binary operations do exactly what addition and subtraction do: They take two objects and produce a new one. We need simpler notation. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. A (binary) operation on a set S is a function ◦ : S × S → S. Binary operations do exactly what addition and subtraction do: They take two objects and produce a new one. We need simpler notation. Definition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Start Abstractly Definition. A (binary) operation on a set S is a function ◦ : S × S → S. Binary operations do exactly what addition and subtraction do: They take two objects and produce a new one. We need simpler notation. Definition. Let S be a set and let ◦ : S × S → S be a binary operation. For all a, b ∈ S we set a ◦ b := ◦(a, b). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Or, in relation notation, for all n ∈ N, the relation + ⊆ (N × N) × N Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Or, in relation notation, for all n ∈ N, the relation + ⊆ (N × N) × N contains the pair (n, 1), n0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Or, in relation notation, for all n ∈ N, the relation + ⊆ (N × N) × N contains the pair (n, 1), n0 , and for all n, m ∈ N for which there is a k ∈ N with (n, m), k ∈ + we have 0 0 that (n, m ), k ∈ +. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Or, in relation notation, for all n ∈ N, the relation + ⊆ (N × N) × N contains the pair (n, 1), n0 , and for all n, m ∈ N for which there is a k ∈ N with (n, m), k ∈ + we have 0 0 that (n, m ), k ∈ +. The relation (which turns out to be a binary operation) is called addition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Adding Natural Numbers Definition. For all m, n ∈ N the relation + : N × N → N is defined by n + 1 := n0 and n + m0 := (n + m)0 . Or, in relation notation, for all n ∈ N, the relation + ⊆ (N × N) × N contains the pair (n, 1), n0 , and for all n, m ∈ N for which there is a k ∈ N with (n, m), k ∈ + we have 0 0 that (n, m ), k ∈ +. The relation (which turns out to be a binary operation) is called addition. (Keep this definition handy. We’ll use it a lot.) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n ∈ N be arbitrary, but fixed. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N : ((n, m), k) ∈ +] . Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. For all m ∈ S we have (n, m0 ), (n + m)0 ∈ +, so m0 ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. For all m ∈ S we have (n, m0 ), (n + m)0 ∈ +, so m0 ∈ S. By the Principle of Induction, we conclude that S = N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. For all m ∈ S we have (n, m0 ), (n + m)0 ∈ +, so m0 ∈ S. By the Principle of Induction, we conclude that S = N. Thus n + m is defined for all m ∈ N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. For all m ∈ S we have (n, m0 ), (n + m)0 ∈ +, so m0 ∈ S. By the Principle of Induction, we conclude that S = N. Thus n + m is defined for all m ∈ N. Because n ∈ N was arbitrary, n + m is defined for all n, m ∈ N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. The relation + is a binary operation on N. Proof. By definition + ⊆ (N × N) × N, nothing more. So we use relation notation in this proof. Our main tool will be the Principle of Induction. Step 1: The relation + is totally defined. Let n∈ N be arbitrary, but fixed. Let S := m ∈ N : [∃k ∈ N :((n, m), k) ∈ +] . By definition, (n, 1), n0 ∈ +, so 1 ∈ S. For all m ∈ S we have (n, m0 ), (n + m)0 ∈ +, so m0 ∈ S. By the Principle of Induction, we conclude that S = N. Thus n + m is defined for all m ∈ N. Because n ∈ N was arbitrary, n + m is defined for all n, m ∈ N. Hence + is totally defined. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor of any natural number Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor of any natural number, we obtain k = n0 = l. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor of any natural number, we obtain k = n0 = l. So 1 ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor k = n0 = of any natural number, we obtain l. So 1 ∈ S. 0 0 Now, if m ∈ S, and (n, m ), k , (n, m ), l ∈ + Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor k = n0 = of any natural number, we obtain l. So 1 ∈ S. 0 0 Now, if m ∈ S, and (n, m ), k , (n, m ), l ∈ +, then k = (n + m)0 = l and therefore m0 ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor k = n0 = of any natural number, we obtain l. So 1 ∈ S. 0 0 Now, if m ∈ S, and (n, m ), k , (n, m ), l ∈ +, then k = (n + m)0 = l and therefore m0 ∈ S. By the Principle of Induction, S = N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor k = n0 = of any natural number, we obtain l. So 1 ∈ S. 0 0 Now, if m ∈ S, and (n, m ), k , (n, m ), l ∈ +, then k = (n + m)0 = l and therefore m0 ∈ S. By the Principle of Induction, S = N. Thus n + m is unique for all n, m ∈ N. We conclude that + is well-defined. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Step 2: The relation + is well-defined. Let n ∈ N be arbitrary, but fixed. Let S := m ∈ N : [∀k, l ∈ N : ((n, m), k), ((n, m), l) ∈ + ⇒ k = l] . If (n, 1), k), ((n, 1), l ∈ +, then, because 1 is not the successor k = n0 = of any natural number, we obtain l. So 1 ∈ S. 0 0 Now, if m ∈ S, and (n, m ), k , (n, m ), l ∈ +, then k = (n + m)0 = l and therefore m0 ∈ S. By the Principle of Induction, S = N. Thus n + m is unique for all n, m ∈ N. We conclude that + is well-defined. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property and then we prove that that set is N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property and then we prove that that set is N. So, if you already know induction, this approach really is not that different. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property and then we prove that that set is N. So, if you already know induction, this approach really is not that different. We’ll get back to “the usual way to do induction” in a little while. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property and then we prove that that set is N. So, if you already know induction, this approach really is not that different. We’ll get back to “the usual way to do induction” in a little while. 2. The idea that a function needs to be proved to be well-defined takes some time getting used to. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Notes 1. To use the Principle of Induction as stated in the Peano Axioms, we define a set that contains all the elements with a certain property and then we prove that that set is N. So, if you already know induction, this approach really is not that different. We’ll get back to “the usual way to do induction” in a little while. 2. The idea that a function needs to be proved to be well-defined takes some time getting used to. Arithmetic modulo m will give us a simpler and pretty natural context. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. 1. Addition is associative Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. 1. Addition is associative, that is, for all n, m, k ∈ N we have that (n + m) + k = n + (m + k). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. 1. Addition is associative, that is, for all n, m, k ∈ N we have that (n + m) + k = n + (m + k). 2. Addition is commutative Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. 1. Addition is associative, that is, for all n, m, k ∈ N we have that (n + m) + k = n + (m + k). 2. Addition is commutative, that is, for all n, m ∈ N we have that n + m = m + n. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proposition. Properties of the addition operation on N. 1. Addition is associative, that is, for all n, m, k ∈ N we have that (n + m) + k = n + (m + k). 2. Addition is commutative, that is, for all n, m ∈ N we have that n + m = m + n. (We need to get used to the vocabulary.) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove associativity, let n, m ∈ N be arbitrary, but fixed. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because (n + m) + k0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because (n + m) + k0 = Bernd Schröder Arithmetic of Natural Numbers (n + m) + k 0 logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because (n + m) + k0 = Bernd Schröder Arithmetic of Natural Numbers 0 0 (n + m) + k = n + (m + k) logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because 0 0 (n + m) + k0 = (n + m) + k = n + (m + k) = n + (m + k)0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because 0 0 (n + m) + k0 = (n + m) + k = n + (m + k) = n + (m + k)0 = n + (m + k0 ). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because 0 0 (n + m) + k0 = (n + m) + k = n + (m + k) = n + (m + k)0 = n + (m + k0 ). By the Principle of Induction, S = N Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because 0 0 (n + m) + k0 = (n + m) + k = n + (m + k) = n + (m + k)0 = n + (m + k0 ). By the Principle of Induction, S = N, so (n + m) + k = n + (m + k) for all k ∈ N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof. We must prove associativity and commutativity, and we will use the Principle of Induction. To prove but fixed. Let associativity, let n, m ∈ N be arbitrary, S := k ∈ N : (n + m) + k = n + (m + k) . 1 ∈ S, because (n + m) + 1 = (n + m)0 = n + m0 = n + (m + 1). Now let k ∈ S. Then k0 ∈ S, because 0 0 (n + m) + k0 = (n + m) + k = n + (m + k) = n + (m + k)0 = n + (m + k0 ). By the Principle of Induction, S = N, so (n + m) + k = n + (m + k) for all k ∈ N. Because n and m were arbitrary, + is associative. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) = 1 + k0 , Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) = 1 + k0 , so k0 ∈ T. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) = 1 + k0 , so k0 ∈ T. Hence T = N Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) = 1 + k0 , so k0 ∈ T. Hence T = N, that is, k + 1 = 1 + k for all k ∈ N. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). To prove commutativity, let n ∈ N be arbitrary, but fixed. Let S := {m ∈ N : n + m = m + n}. To prove that 1 ∈ S, we will perform an induction inside an induction. Let T := {k ∈ N : k + 1 = 1 + k}. Trivially, 1 ∈ T. Now let k ∈ T. Then k0 + 1 = (k + 1) + 1 = (1 + k) + 1 = 1 + (k + 1) = 1 + k0 , so k0 ∈ T. Hence T = N, that is, k + 1 = 1 + k for all k ∈ N. (Back to the “main induction”.) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n = m0 + n, Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n = m0 + n, that is, m0 ∈ S. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n = m0 + n, that is, m0 ∈ S. Thus S = N, Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n = m0 + n, that is, m0 ∈ S. Thus S = N, and for all m ∈ N we have that n + m = m + n. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Proof (cont.). Using k := n we obtain n + 1 = 1 + n, and hence 1 ∈ S = {m ∈ N : n + m = m + n}. Now let m ∈ S. Then n + m0 = n + (m + 1) = (n + m) + 1 = (m + n) + 1 = m + (n + 1) = m + (1 + n) = (m + 1) + n = m0 + n, that is, m0 ∈ S. Thus S = N, and for all m ∈ N we have that n + m = m + n. Because n ∈ N was arbitrary, this establishes commutativity. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 := 1 + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 := 1 + 1 3 := 2 + 1 = (1 + 1) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 := 1 + 1 3 := 2 + 1 = (1 + 1) + 1 4 := 3 + 1 = ((1 + 1) + 1) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 3 4 5 Bernd Schröder Arithmetic of Natural Numbers := := := := 1+1 2 + 1 = (1 + 1) + 1 3 + 1 = ((1 + 1) + 1) + 1 4 + 1 = (((1 + 1) + 1) + 1) + 1 logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 3 4 5 := := := := 1+1 2 + 1 = (1 + 1) + 1 3 + 1 = ((1 + 1) + 1) + 1 4 + 1 = (((1 + 1) + 1) + 1) + 1 and so on, in the usual fashion. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 3 4 5 := := := := 1+1 2 + 1 = (1 + 1) + 1 3 + 1 = ((1 + 1) + 1) + 1 4 + 1 = (((1 + 1) + 1) + 1) + 1 and so on, in the usual fashion. Yes, numerals are merely symbols. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Back to Earth: The Usual Representation of Natural Numbers Definition. We define 2 3 4 5 := := := := 1+1 2 + 1 = (1 + 1) + 1 3 + 1 = ((1 + 1) + 1) + 1 4 + 1 = (((1 + 1) + 1) + 1) + 1 and so on, in the usual fashion. Yes, numerals are merely symbols. Think of roman numerals if that feels “wrong”. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4+3 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4 + 3 = 4 + (2 + 1) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4 + 3 = 4 + (2 + 1) = (4 + 2) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4 + 3 = 4 + (2 + 1) = (4 + 2) + 1 = 4 + (1 + 1) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4 + 3 = 4 + (2 + 1) = (4 + 2) + 1 = 4 + (1 + 1) + 1 = (4 + 1) + 1 + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4 + 3 = 4 + (2 + 1) = (4 + 2) + 1 = 4 + (1 + 1) + 1 = (4 + 1) + 1 + 1 = (5 + 1) + 1 Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4+3 = = = = = = Bernd Schröder Arithmetic of Natural Numbers 4 + (2 + 1) (4 + 2) + 1 4 + (1 + 1) + 1 (4 + 1) + 1 + 1 (5 + 1) + 1 6+1 logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4+3 = = = = = = = Bernd Schröder Arithmetic of Natural Numbers 4 + (2 + 1) (4 + 2) + 1 4 + (1 + 1) + 1 (4 + 1) + 1 + 1 (5 + 1) + 1 6+1 7 logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4+3 = = = = = = = 4 + (2 + 1) (4 + 2) + 1 4 + (1 + 1) + 1 (4 + 1) + 1 + 1 (5 + 1) + 1 6+1 7 If we do too much of this, people will believe we are nuts. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication We Can’t Prove 1 + 1 = 2, But We Can Prove 4+3 = 7 4+3 = = = = = = = 4 + (2 + 1) (4 + 2) + 1 4 + (1 + 1) + 1 (4 + 1) + 1 + 1 (5 + 1) + 1 6+1 7 If we do too much of this, people will believe we are nuts. But, this is a good exercise in mathematical reasoning nonetheless. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. For all m, n ∈ N the relation · : N × N → N is defined by n · 1 := n and n · m0 := n · m + n. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. For all m, n ∈ N the relation · : N × N → N is defined by n · 1 := n and n · m0 := n · m + n. Or, in relation notation, for all n ∈ N, the relation · ⊆ (N × N) × N Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. For all m, n ∈ N the relation · : N × N → N is defined by n · 1 := n and n · m0 := n · m + n. Or, in relation notation, for all n∈ N, the relation · ⊆ (N × N) × N contains the pair (n, 1), n Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. For all m, n ∈ N the relation · : N × N → N is defined by n · 1 := n and n · m0 := n · m + n. Or, in relation notation, for all n∈ N, the relation · ⊆ (N × N) × N contains there the pair (n, 1), n , and for all n, m ∈ N for which is a 0 k ∈ N with (n, m), k ∈ · we have that (n, m ), k + n ∈ ·. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Multiplying Natural Numbers Definition. For all m, n ∈ N the relation · : N × N → N is defined by n · 1 := n and n · m0 := n · m + n. Or, in relation notation, for all n∈ N, the relation · ⊆ (N × N) × N contains there the pair (n, 1), n , and for all n, m ∈ N for which is a 0 k ∈ N with (n, m), k ∈ · we have that (n, m ), k + n ∈ ·. The relation (which turns out to be a binary operation) is called multiplication. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. 3. Multiplication is associative, that is, for all n, m, k ∈ N we have that (n · m) · k = n · (m · k). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. 3. Multiplication is associative, that is, for all n, m, k ∈ N we have that (n · m) · k = n · (m · k). 4. Multiplication is commutative, that is, for all n, m ∈ N we have that n · m = m · n. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. 3. Multiplication is associative, that is, for all n, m, k ∈ N we have that (n · m) · k = n · (m · k). 4. Multiplication is commutative, that is, for all n, m ∈ N we have that n · m = m · n. 5. The number 1 is a neutral element or identity element with respect to multiplication, that is, for all n ∈ N we have that n · 1 = 1 · n = n. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. 3. Multiplication is associative, that is, for all n, m, k ∈ N we have that (n · m) · k = n · (m · k). 4. Multiplication is commutative, that is, for all n, m ∈ N we have that n · m = m · n. 5. The number 1 is a neutral element or identity element with respect to multiplication, that is, for all n ∈ N we have that n · 1 = 1 · n = n. 6. Multiplication is right distributive over addition. That is, for all n, m, k ∈ N we have (n + m) · k = n · k + m · k. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Properties of Multiplication 1. Yes, it’s a binary operation. 2. Multiplication is an abbreviation for repeated addition of a number to itself. 3. Multiplication is associative, that is, for all n, m, k ∈ N we have that (n · m) · k = n · (m · k). 4. Multiplication is commutative, that is, for all n, m ∈ N we have that n · m = m · n. 5. The number 1 is a neutral element or identity element with respect to multiplication, that is, for all n ∈ N we have that n · 1 = 1 · n = n. 6. Multiplication is right distributive over addition. That is, for all n, m, k ∈ N we have (n + m) · k = n · k + m · k. The proof of the last four is a bit tricky. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Then n(m + k) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Then n(m + k) = (m + k) · n Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Then n(m + k) = (m + k) · n = mn + kn Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Then n(m + k) = (m + k) · n = mn + kn = nm + nk Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication Using Properties Shortens Proofs Proposition. Multiplication is left distributive over addition. That is, for all n, m, k ∈ N we have n · (m + k) = n · m + n · k. Proof. Let m, n, k ∈ N be arbitrary, but fixed. Then n(m + k) = (m + k) · n = mn + kn = nm + nk Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd = ac + (bc + ad) + bd Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd = ac + (bc + ad) + bd = ac + (ad + bc) + bd Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd = ac + (bc + ad) + bd = ac + (ad + bc) + bd = (ac + ad) + bc + bd Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd = ac + (bc + ad) + bd = ac + (ad + bc) + bd = (ac + ad) + bc + bd = (ac + ad) + (bc + bd) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science Introduction Addition Properties of Addition Numerals Multiplication FOIL Proposition. Let a, b, c, d ∈ N. Then (a + b)(c + d) = (ac + ad) + (bc + bd). Proof. (a + b)(c + d) = (a + b)c + (a + b)d = (ac + bc) + (ad + bd) = (ac + bc) + ad + bd = ac + (bc + ad) + bd = ac + (ad + bc) + bd = (ac + ad) + bc + bd = (ac + ad) + (bc + bd) Bernd Schröder Arithmetic of Natural Numbers logo1 Louisiana Tech University, College of Engineering and Science

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