PROPERTIES OF NUMBERS
ALGEBRA-2 NOTES-ANSWERS
1)
a+b=b+a
Commutative property of addition
2)
m+n=n+m
Commutative property of addition
3)
a·b=b·a
Commutative property of multiplication
4)
a + b + c = (a + b) + c = a + (b + c)
Associative property of addition
5)
(m + n) + p = m + (n + p)
Associative property of addition
6)
a · (m + n) = a · m + a · n
Distributive property
7)
a+0=a=0+a
8)
M·1=1·M=M
9)
a + (– a) = 0 = (– a) + a
Identity property of addition (Existence of neutral element for
addition)
Identity property of multiplication (Existence of neutral element
for multiplication)
Inverse property of addition (The result is the neutral element)
10)
p+0=p
Identity property of addition
11)
5 + (– 5) = 0
Inverse property of addition
12)
7=7
Reflexive property
13)
a = b, then b = a
Symmetric property
14)
a = b, and b = c, then a = c
Transitive property
15)
Substitution property
16)
If a = b, then a may be replaced by b
and b may be replaced by a
If a = b, then a + k = b + k
17)
If a = b, then a – k = b – k
Subtraction property of equality
18)
If a = b, then a · k = b · k
Multiplication property of equality
19)
If a = b, then a/k = b/k, k ≠ 0
Division property of equality
Addition property of equality
*20) (A + B ) + C = (B + A) + C
Commutative property of addition
*21) (A + B ) + C = C + (A + B)
Commutative property of addition
*22) A · ( M + N) = (M + N) · A
Commutative property of multiplication
*23) A + B = C + D, then C + D = A + B
Symmetric property
*24) (M + N) + 0 = (M + N)
Identity property of addition
*25) For any two real numbers, a and b,
exactly one of the following statements
is true: a < b
a=b
a>b
Trichotomy property
The Closure Property
We need to have a set of numbers and an operation to state this property.
Example-1: Write a valid argument to show that the
set of natural numbers is closed for addition or use a
counterexample to show that it is not.
Example-3: Write a valid argument to show that the
set of real numbers (R) is closed for multiplication
or use a counterexample to show that it is not.
Let the set of natural numbers (N) be the domain for
the arithmetic operation
Let the addition be the arithmetic operation
Let any two natural numbers be 6 and 2
When we add them, we have: 6 + 2 = 8
Is the answer a natural number? _YES_
Let the set of real numbers (R) be the domain for
the arithmetic operation
Let the multiplication be the arithmetic operation
Let any two real numbers be 4.5 and 2
When we multiply them, we have: (4.5)(2) = 9
Is the answer a real number? YES
Is the addition of any two natural numbers always a
natural number? YES
Is the multiplication of any two real numbers
always a real number? YES
When two natural numbers are added, the result will
always be a natural number.
When any two real numbers are multiplied, the
result will always be a real number.
The set of real numbers IS closed for multiplication.
There are no counterexamples.
The set of natural numbers IS closed for addition.
There are no counterexamples.
YOUR TURN:
Example-2: Write a valid argument to show that the
set of whole numbers (W) is closed for subtraction
or use a counterexample to show that it is not.
1. Write a valid argument to show that the set of
Natural numbers (N) is closed for multiplication or
use a counterexample to show that it is not.
Let the set of whole numbers (W) be the domain for
the arithmetic operation
Let the subtraction be the arithmetic operation
Let any two whole numbers be 5 and 3
When we subtract them, we have: 5 – 3 = 2
Is the answer a whole number? YES
2. Write a valid argument to show that the set of
integers (Z) is closed for division or use a
counterexample to show that it is not.
Let’s reverse the order: 3 – 5 = – 2
Is the answer a whole number? NO
This is a counterexample.
The set of whole numbers IS NOT closed for
subtraction.
3. Write a valid argument to show that the set of real
numbers (R) is closed for square root or use a
counterexample to show that it is not.
4. Write a valid argument to show that the set of
rational numbers (Q) is closed for division or use a
counterexample to show that it is not.
5. Write a valid argument to show that the set of real
numbers (R) is closed for cube root or use a
counterexample to show that it is not.