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Sets of numbers

In the world of mathematics we have categorized all the numbers that exist into certain sets.  Let’s describe the certain sets that exist as well as their properties:

The Set of Natural Numbers

Definition:  The set \(\mathbb{N}\) of natural numbers is defined by

\(\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)

The set of natural numbers is often referred to as the set of counting numbers, because they are those numbers that we use to count.  Notice the three dots in the set notation.  That means that the set continues without end.  Also notice that 0 is not included in the natural numbers.

The Set of Integers

Definition:  The set \(\mathbb{Z}\) of integers is defined by

\(\mathbb{Z} = \left\{ { \ldots , - 3, - 2, - 1,0,1,2,3, \ldots } \right\}\)

The set of integers contains all whole numbers (including 0) and their opposites.  Notice that the set of natural numbers is contained within the set of integers.

The Set of Rational Numbers

Definition:  The set \(\mathbb{Q}\) of rational numbers is defined by

\(\mathbb{Q} = \large \left\{ {\frac{m}{n}:m,n{\text{ are integers}}} \right\}\)

Here is the set of rational numbers, all those numbers that can be expressed as a ratio of two integers.  Furthermore, the set of rational numbers includes all those numbers whose decimal representation terminates or repeats.

The Set of Irrational Numbers

Definition:  The set of Irrational Numbers is defined by those numbers whose decimal representations never terminates or repeats.  Some famous irrational numbers include \(\pi \) and \(\sqrt 2 \). In fact, \(\sqrt n \) is irrational if \(n\) is not a perfect square.

Together all these sets combined make up the SET OF REAL NUMBERS.

 

Example:  Name the set or sets to which 0 belongs.

Solution:  By inspection we see that 0 is a member of the integers, the rational numbers (since \(\large \frac{0}{1} = 0\), and the real numbers.

 

Example:  Name the set or sets to which \(\sqrt {82} \) belongs.

Solution:  Since 82 is not a perfect square, \(\sqrt {82} \) is irrational.  Therefore it cannot be rational, an integer, or a natural number.  But it is a real number.

 

Below you can download some free math worksheets and practice.


Downloads:
4598 x

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Beginning-Algebra-Sets-of-numbers-easy

Watch below how to solve this example:

 

Downloads:
2003 x

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Beginning-Algebra-Sets-of-numbers-medium

Watch below how to solve this example:

 

Downloads:
1721 x

Name the set or sets to which each number belongs.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Beginning-Algebra-Sets-of-numbers-hard

Watch below how to solve this example:

 
 
 

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Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
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Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
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Simplifying and excluded values
Systems of Equations and Inequalities
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Solving by elimination
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