Real Numbers Week 1 Topic 1. Real Numbers Irrational Numbers Numbers that cannot be written as a fraction √2, π Rational Numbers Numbers that can be written.

Real NumbersWeek 1 Topic 1

Real Numbers

Irrational NumbersNumbers that cannot be written as a fraction

√2, π

Rational NumbersNumbers that can be written as a fraction

Decimals that repeat

Decimals that stop

√25, ½, 5, 0.123, 0.333333…

Real NumbersSet of all irrational and rational numbers

Real Numbers

IntegersPositive and negative counting numbers (plus 0){…-3, -2, -1, 0, 1, 2, 3…)

Whole NumbersCounting numbers starting at 0{0, 1, 2, 3…}

Natural NumbersCounting numbers starting at 1{1, 2, 3…}

Real Numbers

Infinite sets- not countableWhole numbers greater than 8

{3, 4, 5 …}

Finite sets- countableIntegers between 2 and 17

{2, 5, 7, 19, 23}

Real Numbers

Estimating the value of an irrational numberCompare perfect square values

List perfect squares close to your value

√67

√49 = 7; √64 = 8; √81 = 9

67 is between 64 and 81 so √67 is between 8 and 9

8 < √67 < 9

Real Numbers

1. Which of the following represents an infinite set of numbers?a. {1/2, 1/3, ¼, 1/5}

b. {Negative integers}

c. {-3, -1, 0, 1, 3}

d. {Natural numbers between 5 and 20}

Real Numbers1. Which of the following represents an infinite

set of numbers?a. {1/2, 1/3, ¼, 1/5}

This set has a clear start and stop, we see exactly 4 values in the set so it is countable or finite

b. {Negative integers}

integers go off to infinite so this set is not countable

c. {-3, -1, 0, 1, 3}

We can count the 5 values in this set.

d. {Natural numbers between 5 and 20}

We can list and count the values in this set. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Real Numbers

2. Which of the following is an irrational number?

a. √5

b. √9

c. 7

d. 3.78

Real Numbers

2. Which of the following is an irrational number?

a. √5

b. √9 = 3 whole numbers are rational

c. 7 = 7/1 whole numbers are rational

d. 3.78 = 378/100 decimals that stop are rational

Real Numbers

3. Between which two consecutive integers is √113 ?

a. 12 and 13

b. 8 and 9

c. 10 and 11

d. 11 and 12

Real Numbers3. Between which two consecutive integers is

√113 ?

a. 12 and 13

b. 8 and 9

c. 10 and 11

d. 11 and 12

82 = 64; 92 = 81; 102 = 100; 112 = 121; 122 = 144; 132 = 169

Number PropertiesWeek 1 topic 2

Number Properties

Number Properties Rap

Math Properties

Number Properties

Commutative PropertyNumbers can be added or multiplied in any order.

1 + 2 = 2 + 12(3) = 3(2)

Associative PropertyWhen adding, changing the grouping doesn’t matter.

(1 + 2) + 3 = 1 + (2 + 3)

When multiplying, changing the grouping doesn’t matter.

2(3x4) = (2x3)4

Number Properties

IdentityAdding 0 doesn’t change a value

Multiplying by 1 doesn’t change the value

InverseAdding the opposite gives you 0

Multiplying by the reciprocal gives you 1

Distributive Property3(a + b) = 3a + 3b

Number Properties

ClosureWhen you add or multiple real numbers together the answer will also be a real number.

Number Properties

Number Properties

When we multiply by 1 the number keeps its value or “identity”.

Number Properties

Number Properties

This is the Closure Property

Number Properties

Number Properties

Number Properties

Number Properties

The numbers are being regrouped so this is the associative property.

Number Properties

Number Properties

The multiplicative inverse is the reciprocal. We use it to make a number turn into 1.

Integers and Absolute ValuesWeek 1 Topic 3

Integers

Adding two positive integers

Just add.

Answer will be a positive

Adding a positive and a negative

Subtract

Answer will be the same as the larger of the two numbers

Adding two negatives

Just add

Answer will be negative

Absolute Value

Absolute Value is the distance a number is from zero on the number line.

|-2| = 2

|3 – 6| = |-3| = 3

Order of OperationsWeek 1 Topic 4

Order of Operations

Order of Operations Rap

Order of ops rap 2

Order of Operations

Order of Operations

Parenthesis 22 – 2[5 + 3(5)]

Brackets (more parenthesis) 22 – 2[5 + 15]

22 – 2[20]

Multiplication 22 – 40

Subtraction -18

Order of Operations

Order of Operations

2[7 + 5(-3)]

2[7 + (-15)]

2(-8)

-16

Order of Operations

Order of Operations

2(-48 / 4 x 3)

2(-12 x 3)

2(-36)

-72This one is tricky…we have to multiply and divide at the same time from left to right.

Scientific NotationWeek 1 Topic 5

Scientific Notation

A number written as a product of a power of 10 and a decimal number greater than or equal to 1 and less than 10.

3.72 x 106

When adding and subtracting the exponents must be the same…or we have to rewrite them in standard form first.

3.72 x 106 + 1.5 x 106 = (3.72 + 1.5) x 106 = 5.22 x 106

Scientific Notation

MultiplyingMultiply the factors, add the exponents

DividingDivide the factors, subtract the exponents

Scientific Notation

Scientific Notation

Since the exponents have the same value we can add the factors 7.8 and -4.2.

(We end subtracting)

7.8 – 4.2 = 3.6

So our answer is 3.6 x 1020

Scientific Notation

Scientific Notation

5.1 / 1.7 = 3

-6 – (-4) = -6 + 4 = -2

3 x 10-2

Scientific Notation

Scientific Notation

Asia / Australia

(1.72 x 107) / (3.13 x 106)

1.7 is about half as big as 3.13

1.72/3.13 ≈ .55

Subtract the exponents… 7 – 6 = 1

.55 x 101 = 5.5