1.1 ­ Real Numbers.notebook
September 14, 2020
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1.1 - Real Numbers
Skills:
Properties of Real Numbers
Addition and Subtraction Properties
Multiplication and Division Properties
The Real Number Line
Sets, Intervals and Set Notation
Absolute Value and Distance
When we dive into any deeper mathematics, its important that we have an
established set of ground rules that allow us to manipulate numbers, in this
section, we'll review most of these properties that you've seen over the years!
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1.1 ­ Real Numbers.notebook
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The Set of Real Numbers
The set of Real Numbers is made up of the subsets of Rational and Irrational Numbers .
The set of Rational Numbers includes the subsets of Integers, Whole Numbers, and Natural Numbers.
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Sets of Numbers
Natural Numbers are the counting numbers: 1, 2, 3, 4, 5...
Whole Numbers are the counting numbers plus zero: 0, 1, 3, 4...
Integers include the whole numbers plus negatives: ... -2, -1, 0, 1, 2...
Rational Numbers include Integers as well as fractions/decimals.
Irrational Numbers are the set of numbers that can NOT be expressed as
fractions. This includes numbers like π, √2, e, √7.
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Properties of Real Number
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Addition and Subtraction Properties
In Addition and Subtraction zero is a special number referred to as the
additive identity. This is because for any real number 'a', a + 0 = a.
Subtraction is the operation that undoes addition, and by definition we say:
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Properties of Negative Numbers
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Multiplication and Division Properties
In multiplication, the number 1 is the multiplicitive identity because for any
real number 'a', a(1) = a.
For every non-zero number 'a', there exists an inverse
times it's inverse is equal to 1.
so that a number
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Properties of Fractions
Given two real numbers 'a' and 'b', we can arrange them as a fraction such
that:
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The Real Number Line
All real numbers can be arranged by their value on a number line.
When comparing numbers we use the operators <, > or = to compare two
numbers.
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1.1 ­ Real Numbers.notebook
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Sets and Intervals
When we talk about sets in mathematics, we are talking about a group of
numbers. The set of odd numbers, the set of multiples of 5, the set of
negative numbers, etc.
The individual numbers in the sets are called elements.
Sets are usually designated with capital letters, while elements are written with
lower case letters.
Example
is read as "a is an element of set B"
is read as "b is not an element of set S"
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Set Builder Notation
We can define as set as:
elements 1, 3, 5, and 7.
. This is the set S, that contains the
We could also define this as:
"The Set S is defined as x, such that x is made up of odd numbers 1 through 7"
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Union and Intersection of Sets
Given: A = {1, 2, 3} B = {2, 3, 5} and C = {1, 10, 100}
The Union of two sets combines the elements in those sets.
=
=
The Intersection of two sets asks what those sets have in common.
=
=
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Try This:
Given the sets: H = {0, 5, 10, 15}, I = {1, 3, 5, 7} and J = {10, 20, 30}
Find:
H
I
I
J
J
H
I
H
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Intervals
An Interval describes a set of numbers in a row . Intervals can be described
as open or closed. An open interval means the numbers at the end of the
interval are NOT in the set, and a closed interval means the numbers at the
end of the interval are in the set . An interval can be open on one sided and
closed on another.
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Absolute Value and Distance
Absolute value is defined as the distance a number is from zero on the number
line. Distance is always positive or zero. It can be defined as the piecewise
function:
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