Notes
Real Numbers - §1.1 - 1.3
Fall 2013 - Math 1010
i to e: ”You’re irrational.” e to i: ”Get real.”
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Roadmap
Notes
For §1.1:
I
Understand a set, namely the set of real numbers and its subsets.
I
Order of real numbers.
I
Distance between two real numbers.
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The Real Numbers Set
Notes
A set is a collection of objects.
Its notation uses brackets, {}, with the objects in between, or a single
symbol or letter, or an interval.
The set used for arithmetic is the set of real numbers. Its has many
notations: ”R”, ”(−∞, ∞)”, and a number line.
The real number has many notable subsets. They include:
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”N”, the natural numbers
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The whole numbers
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”Z”, the integers
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”Q”, the rational numbers
I
The irrational numbers
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Distance between two real numbers
Notes
The distance between two real numbers, when a ≤ b, is the value b − a.
The distance between a number and 0 is its absolute value.
This is denoted by vertical bars, | |.
|a| =
(
a
−a
a≥0
a<0
The opposite of a number a is −a.
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Roadmap
Notes
For §1.2:
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Operations in R: +, −, ×, ÷.
I
Exponents
I
Order of Operations
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Tips - Arithmetic operations.
Notes
I
Real numbers involved in addition or subtraction are called terms.
I
When adding or subtracting fractions, make each fraction have a
common denominator.
I
Real numbers involved in multiplications are called factors.
I
When multiplying fractions, factor the numerators and denominators.
Then divide out any common factors.
I
When dividing a ÷ b, a is the dividend and b is the divisor.
I
When dividing fractions, replace the divisor with its reciprocal. Then
multiply the fractions.
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Tips - Exponents / Powers
Notes
I
an can be written as a · a · a · · · a, with n factors of a.
I
For a > 0, (−a)n is less than zero when a is odd. Otherwise (−a)n is
positive when n is even.
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(−a)n and −an can be different.
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Order of Operations
Notes
To evaluate an expression involving more than one operation, use the
following order.
1. First do operations that occur within symbols of grouping.
(Symbols of grouping include parentheses and brackets.)
2. Then evalaute powers.
3. Then do multiplications and divisions from left to right.
4. Finally, do additions and subtractions from left to right.
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Roadmap
Notes
For §1.3:
I
Use properties of R.
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Properties of addition
Notes
Reminder: Using opposites as terms is the same as subtraction.
Commutative:
a+b =b+a
Associative:
(a + b) + c = a + (b + c)
Additive identitiy (zero):
a+0=0+a=a
Additive inverse (opposite):
a + (−a) = 0
M 1010 §1.1 - 1.3
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Properties of Multiplication
Notes
Reminder: For division, you can change divisors to their reciprocals and
then use multiplication.
Commutative:
same as a · b = b · a
ab = ba
Associative:
(ab)c = a(bc)
Multiplicative identity (one):
a·1=1·a=a
Mutilplicative inverse (reciprocal):
a·
(Math 1010)
1
= 1, when a 6= 0
a
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Distributive property
Notes
a(b + c) = ab + ac
(a + b)c = ac + bc
a(b − c) = ab − ac
(a − b)c = ac − bc
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Cancellation properties
Notes
Properties of real numbers in practice are less formal than a list. What are
the formal steps used in cancelling terms and factors when solving this
equation for x?
7x + 9 = −5x − 15
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Assignment
Notes
Assignment:
For Wednesday:
1. Exercises from §1.1 - 1.3 due Wednesday, September 4
2. Clicker registration is due by Wednesday. September 4
Vocabulary: real numbers, natural numbers, integers, rational numbers,
irrational numbers, absolute value, opposite, numerator, denominator,
dividend, divisor, reciprocal, associative, commutative, identity,
cancellation
Understand: Identify the type of a real number. Find the distance
between two real numbers. Change subtraction to addition using
opposites, and change division into multiplication using reciprocals.
Memorize and practice the order of operations. Gain skill recognizing and
using the properties of real numbers.
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Notes