Real Number Systems Subsets

The Number System

by Susan Joyce

​The Big Category: Complex Numbers

• ​1. All numbers can be expressed as a complex number

• ​2. Complex numbers have a real​part and an imaginary part (containing the letter i, which represents the square root of negative 1)

​Complex Numbers

​1. You will study complex numbers later in math

​2. For now, be able to recognize a number written as a complex number

​3. If I wanted to write the number "2" as a complex number, I would write "2 + 0i" where 2 is the real part and 0*i (=0) is the imaginary part

​Complex Numbers

• ​If I wanted to write an imaginary number (with an i) as a complex number, I would write 0 + 5i

• ​ALL NUMBERS FALL INTO THE BIG CATEGORY OF COMPLEX NUMBERS

• ​Complex numbers are divided into two subsets: REAL AND IMAGINARY

Multiple Choice

All numbers can be written as a complex number.

(True or False)

• True

• False

Multiple Choice

All complex numbers contain a real part and an imaginary part.

(True or False)

• True

• False

Multiple Choice

The number 5 is not a complex number.

• True

• False

Multiple Choice

6i is the real part of the complex number 8 + 6i

(True or False)

• True

• False

​Complex Numbers Subsets

​1. Complex numbers are divided into two groups: real and imaginary.

​2. Imaginary numbers are the square roots of a negative numbers

​like

​3. Real numbers are divided into two sets: Rational and Irrational

​

​What do we have so far?

• ​1. Complex Numbers (a + bi)

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• ​2. Complex Numbers are divided into 2 groups: Real and Imaginary

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• ​Imaginary Numbers do not have subsets (or sub categories)

​Real Number Subgroups

• ​Real Numbers are divided into two sets: Rational and Irrational

• ​

• ​Rational Numbers can be expressed as the ratio of two numbers: Fractions, Terminating Decimals, Repeating Decimals

​Real Numbers: Rational and Irrational

• ​Irrational numbers are numbers with non-terminating, non-repeating decimals, like square roots of numbers that aren't perfect squares

• ​

​Think of it this way: A rational person has a meaningful conversation. An irrational person talks on and on and on.

​Irrational Numbers

• ​Non-terminating decimals or numbers that cannot be represented as a simple fraction

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• ​Square roots of numbers that are not perfect squares

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• ​Doesn't have any other subsets

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​Rational Numbers

• ​Numbers that can be expressed as a simple fraction

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• ​Terminating decimals or repeating decimals

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• ​Divided into subsets: Integers, Whole Numbers and Natural Numbers

​Integers

• ​Positive and Negative Whole Numbers and 0

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• ​An integer is a rational number because it can always be expressed as a number with 1 as the denominator: 4/1, -5/1, 1001/1

​Whole Numbers

• ​0 and all the natural numbers

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• ​A negative number is an integer, but NOT a whole number

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• ​An whole number is a number that the numerator is a multiple of the denominator. It has no frational parts.

​Natural Numbers

• ​Also called counting numbers

• ​

• ​Does not include 0 or negative numbers

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​From the inside to the outside

• ​Natural Numbers: Positive whole numbers

• ​Whole Numbers: Natural numbers and 0

• ​Integers: Positive and Negative Whole Numbers

• ​Rational: Integers and Fractions, Terminating Decimals and Repeating Decimals

​From the inside to the outside(con't.)

• ​Irrational Numbers: Numbers that cannot be expressed as frations; Square roots of numbers not a perfect square

• ​

• ​Irrational Numbers and Rational Numbers (Natural Numbers, Whole Numbers, Integers and Fractions) make up the REAL number system

​From the inside to the outside (con't)

• ​Imaginary Numbers: Square roots of negative numbers

• ​

  ​REAL and IMAGINARY numbers make up the COMPLEX number system

​A Couple of Things

• ​All natural numbers are also real numbers, but not all real numbers are natural numbers (no fractions, negative numbers or zero)

• ​All whole numbers are also real numbers, but not all real numbers are whole numbers (no fractions, negative numbers)

​A Couple of Things

• ​All integers are real numbers, but not all real numbers are integers (no fractions)

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• ​Every number is a complex number, but not all complex numbers are real numbers (some are imaginary)

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• ​Notice that the groups that start with an "i" don't have subgroups​(just a coincidence)

Multiple Choice

To which category does the number 3 belong?

• All of These

• Integer Only

• Rational # Only

• Whole # Only

Multiple Choice

Say 'TRUE' or 'FALSE' Every irrational number is a real number

• TRUE

• FALSE

Multiple Choice

To which category does -3.4 belong?

• Whole # Only

• Integer Only

• Rational # Only

• Integer & Rational # Only

Multiple Choice

A repeating decimal or decimal that terminates is called a 

• Natural Number

• Irrational Number

• Rational Number

• Whole Number

Multiple Choice

What is true about irrational numbers?

• They are terminating decimals.

• They are non-repeating decimals.

• They are fractions.

• They are natural numbers.

Multiple Choice

A number can be rational and irrational.

• True

• False

Multiple Choice

What is a possible integer that could be between -6 and 2 on a number line?

• 4

• -8

• 3

• -3

Multiple Choice

What are your counting numbers (1,2,3..)?

• Rational Numbers

• Whole Numbers

• Natural Numbers

• Irrational Numbers

Multiple Choice

π or PI is classified as which?

• Rational

• Irrational

• Rational, Real

• Irrational, Real

Multiple Choice

Every rational number is   

• a natural number

•  an integer

•  a real number 

•  a whole number

Multiple Choice

Decimal representation of a rational number cannot be_____

• Terminating

• Non-terminating

• Non-terminating and repeating

• Non-terminating and non-repeating

Closed and Open Sets

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​Closed Sets

• ​A set is closed if you perform an operation on two elements of the set and you get something that is also a member of the set

• ​

• ​In student terms: Integers are closed under addition because if you add two integers together, you get an integer

• 2 + 5 = 7, all of those numbers are integers

​Closed Sets

• ​Integers are closed under multiplication because the product of two integers is an integer

• ​

​Open Sets

• ​The set of irrational numbers is open under multiplication because the product of two irrational numbers could be a rational number.

• ​

Multiple Choice

Is the sum of  $$3\sqrt{2}$$  and  $$4\sqrt{2}$$  rational or irrational? Why? 

• Rational because the sum can be expressed as a fraction. 

• Rational because the sum cannot be expressed as a fraction. 

• Irrational because the sum can be expressed as a fraction. 

• Irrational because the sum cannot be expressed as a fraction.

Multiple Choice

Which of the following is not always true?

• The sum of two rational numbers is rational.

• The product of two irrational numbers is rational.

• The sum of an irrational number and a rational number is irrational.

• The product of a nonzero rational number and an irrational number is irrational.

Multiple Select

Which set(s) does the number 5 classify under? Select all that apply

• Natural

• Whole

• Rational

• Irrational

Multiple Choice

We use the letter Q to represent the set of Rational Numbers because...

• R is already used for Real

• Q stands for quarter of numbers

• Q stands for quotient

• Q stands for quadrant