Irrational Numbers

Rational and Irrational Numbers Review

Algebra 1/ Intermediate Algebra

​What is a Rational Number?

• ​A rational number is any number that can be written as a fraction where the denominator (bottom number) is not ZERO!

  • ​Repeating decimals

  • ​Terminating decimals

​Types of RATIONAL NUMBERS

• ​Natural Numbers

  • ​aka "counting numbers." These are the numbers that you can count (so, basically all positive whole numbers with the exception of 0).

• ​Whole Numbers

  • ​Whole numbers are all natural numbers and 0!

• ​Integers

  • ​Basically all positive and negative WHOLE numbers (and 0).

​What is an irrational number?

• ​An irrational number can not be written as a fraction.

  • ​ALL repeating decimals

  • ​

  • ​Square roots of imperfect squares (aka- square roots that don't work out)

Multiple Select

Which set of numbers does the following number belong?

19736

• Irrational Number

• Natural Number

• Whole Number

• Integer

Multiple Choice

Which set of numbers does the following number belong?

17.8

• Rational Number

• Natural Number

• Whole Number

• Integer

Multiple Select

Which set(s) of numbers does the following number belong?

$$\sqrt[]{49}$$  

(Hint: USE DESMOS)

• Natural Number

• Whole Number

• Integer

• Irrational 

Multiple Choice

Which of the following set(s) of numbers does the following belong?

$$\sqrt[]{99}$$  

*HINT: USE DESMOS!

• Rational Number

• Whole Number

• Irrational Number

• Natural Number

Multiple Choice

Which of the following set(s) of numbers does the following beloing?

$$\frac{19}{4}$$  

*HINT: USE DESMOS!

• Natural Number

• Whole Number

• Integer

• Rational Number

Poll

What type of decimal is this?

0.6

• Terminating

• Repeating

• Non-terminating

• This is not a decimal.

Multiple Choice

What type of decimal is this?

$$\sqrt[]{100}$$  

*HINT: USE DESMOS

*HINT 2: Whole Numbers automatically TERMINATE

• Terminating

• Repeating

• Non-Terminating

• Banana

Multiple Choice

What type of decimal is this?

$$\frac{2}{5}$$  

*HINT: USE DESMOS

• Terminating

• Repeating

• Non-terminating

Multiple Choice

What type of decimal is this?

$$-\frac{2}{3}$$  

*HINT: USE DESMOS

• Terminating

• Repeating

• Non-terminating

Multiple Choice

What type of decimal is this?

0.35217534....

*HINT: The dots at the end means it GOES ON FOR-EV-ER.............

• Repeating

• Terminating

• Non-terminating

Operations with Rational and Irrational Numbers

• ​When you're ADDING:

  • ​Rational+Rational = Rational

  • ​Rational + Irrational= Irrational

  • ​Irrational +Irrational= Irrational

• ​When you're MULTIPLYING:

  • ​Rational x Rational= Rational

  • ​Rational x Irrational = Irrational

  • ​Irrational x Irrational= Irrational or Rational

Multiple Choice

True or False?

A rational number multiplied by an irrational number is irrational.

• True

• False

Multiple Choice

Where's the LIE?!?!?

(Which one of the following is FALSE)

• Natural numbers are also whole numbers.

• Whole numbers are also integers.

• Whole numbers are rational numbers.

• Natural numbers are irrational.

Multiple Choice

Which of the following will give you an irrational answer?

*HINT: Remember the rules for operations with rational/irrational numbers.

*HINT 2: USE DESMOS!!

• $$\sqrt[]{16}+5$$  

• $$\frac{2}{3}+\sqrt[]{144}-3.3$$  

• $$\sqrt[]{5}\times\sqrt[]{20}$$  

• $$3\pi\ +\sqrt[]{5}$$  

Multiple Choice

Choose the correct sign to compare the numbers.

-14 _____ -12

• $$<$$  

• $$>$$  

• =

Multiple Choice

Choose the correct sign to compare the numbers.

$$\left|-5\right|\ \ \ \ \ \ \ \ \ \left|-3\right|$$  

• $$<$$  

• $$>$$  

• $$=$$  

​Let's talk about EXPRESSIONS

​An algebraic expression is a series of numbers, letters and operation signs without an = sign.

​

​We call each part of an expression a term. Terms are separated by + and - signs inside of the expression. Terms can contain coefficients (big numbers), variables (letters), and exponents (little flying numbers).

​More about Expressions

• ​A constant is a number that does NOT have a variable.

• ​The degree of an expression can be found by adding all of the UNLIKE variables' exponents.

Multiple Choice

What is the constant of the following expression?

$$12x^3-8x^2+9x-10$$  

*Hint: Remember to KEEP the sign!

• 12

• -8

• 9

• -10

Multiple Choice

How many terms are in the following expression:

$$12x^3-8x^2+9x-10$$  

*HINT: Terms are SEPARATED by + and - signs.

• 1

• 2

• 3

• 4

Multiple Choice

What is the degree of the following expression:

$$12x^3-8x^2+9x-10$$  

• 3

• 2

• 1

• 0

Multiple Choice

What is the coefficient of the SECOND term in the following expression?

$$12x^3-8x^2+9x-10$$  

• 12

• -8

• 9

• 2

Multiple Choice

What is the coefficient of the FIRST term in the following expression?

$$12x^3-8x^2+9x-10$$  

• 12

• -8

• 9

• -10

Multiple Choice

Evaluate when x=2.

5x+10

• 15

• 20

• 5

• 25

Multiple Choice

Evaluate for x=5.

3x+2

• 3

• 17

• 21

• 13

Multiple Choice

Evaluate for x=2 and y=-1

12x+7y-1

• 16

• 32

• -9

• 0

Multiple Choice

Evaluate for x=3 and y= -3

$$8x^2+2y+8$$  

• 74

• -81

• 33

• 124

Multiple Choice

What is the solution of the following?

4x+12=x+21

*HINT: Use DESMOS!

• 4

• -3

• 3

• 5

Multiple Choice

What is the solution of -7x +2=9

*HINT: USE DESMOS

• -1

• 1

• 2

• 3