Vanessa Graulich

Certified Math Teacher

6th-12th

#### Real numbers are subdivided into integers, rational and irrationals.

Integers are whole numbers that can be negative or positive.

Whole numbers: -2,-1,0,1,2,3....

Even integers: -4,-2,0,4,6,8,10(0 is an even integer)

Odd integers: -3,-1,1,3,5

Rational: All integers and fractions

0.5,14 ,34, etc.

Irrational: These numbers have infinite decimals: π,√2,√3

THE NUMBER ZERO IS AN EVEN INTEGER

#### The right side has positive numbers and the left side has negative numbers.

The closer a negative number is to zero, the greater its value.

Examples:

-5 is greater than -10
-1 is greater than -4

-10 is greater than -20

-0.5 is greater than -1

When you add two numbers with different signs, you subtract and keep the sign of the largest number.
Examples: - 15 +7 = - 8
- 4 + 7 = +3
When you add two numbers with the same sign, you add and keep the same sign.
Examples: -8 – 12= - 20
-6 – 4= - 10
Let’s do another example:
-4 + 6 + 5 – 8 – 12 + 16 + 21=?

Step One: Group the positive numbers and the negative numbers together:

-4-8-12+6+5+16+21=?
Step Two: Do the operation for the negative numbers:
(-4-8-12 = -24)
Step Three: Do the operation for the positive numbers:
(6+5+16+21= 48)
Final Step: Now group the two numbers together-24+48 = 24

Multiplying and Dividing
When you have to multiply or divide with different signs remember these rules:

Examples:
(-4) times (-3) = 12
(-4 ) times ( 3 ) = -12
(-7) divided by (-4 ) = 28
(-7) divided by ( 4 ) = -28

Look at the difference between
-5 -3= -8

and

(-5)(-3)=15

In the first case (-5-3=-8)
we are grouping the numbers together,
the second case (-5)(-3) =15
we are multiplying the two numbers.

Let’s do more examples:
-4+10= 6
(-4)(10)= -40
-5-5= -10
(-5)(-5)= 25

You can NEVER divide by “ZERO”
It is a BIG NO NO in math
The answer is undefined when you divide by zero any number

4/0= undefined !
-5/0= undefined !
6/0= undefined !

YOU NEVER DIVIDE BY ZERO

BUT IF YOU DIVIDE ZERO BY A NUMBER,

0/4= 0
0/-6= 0
0/5=0

Absolute Value
The absolute value of a number is always positive; whatever is inside the absolute value will be positive.
Example: |-4|= 4

-|4|= - 4

In this example, the negative is outside the absolute value.

Therefore, when you find the absolute value of 4 you get positive 4.

Remember, the negative is still there so the answer is -4.
There is a difference between a parenthesis and an absolute value.
For example:
-(-3) = 3 because you can say (-1)(-3) = 3
but if you have -|-3|, the negative outside cannot affect the absolute value, therefore, -|-3|= -3

Order of Operations
Order of operations determines the steps to be taken to solve an expression.

P.E.M.D.A.S: This mnemonic is used to remember the order of operations.

How can you remember P.E.M.D.A.S?
Please Excuse My Dear Aunt Sally

Example:
Simplify the following expression:

Step 1: Parenthesis
9 ÷ 3+ 5•3^2 - 4 + 5^2

Step 2: Exponents:
9 ÷ 3+ 5•9 - 4 + 25

Step 3: Multiplication:
9 ÷ 3+ 45 - 4 + 25
Step 4: Division:
3 + 45 - 4 + 25
73 -4
Step 6: Subtraction:
69

Factors of a number
The factors of a number are the positive integers that evenly divide into that number.
For example, the factors of 8 are 1,2,4 and 8.

Let’s do a couple of examples:
The factors of 12 are 1,2,3,4,6 and 12.
The factors of 40 are 1,2,4,5,8,10,20 and 40.

Divisibility Rules

Let’s go over the basic rules

Let’s do examples how to find the divisibility of different numbers:
The number 612 :
* It is divisible by 2 because the last digit (2) is even.
*It is divisible by 3 because when you add its digits 6+1+2 =9 the total is divisible by 3.
*It is divisible by 4 because its last digits (12) is divisible by 4.
*It is divisible by 6 because it is divisible by 2 and 3.
The number 111:
*It is divisible by 3 because when you add its digits 1+1+1 =3 the total is divisible by 3.
The number 450 :
* It is divisible by 2 because the last digit is even
*It is divisible by 3 because when you add its digits 4+5+0 =9 the total is divisible by 3.
*It is divisible by 5 and 10.
*It is divisible by 6 because it is divisible by 2 and 3.

If they ask you what are the factors of 24, then you have to start with the question.

Can I divide 24 by 1? 2?,3?

and so on until you get to 24.

Let's do it.
Can I divide it by 1?
Yes, then 1 is a factor.
Can I divide it by 2?
Yes, then 2 is a factor.
Can I divide it by 3?
Yes, then 3 is a factor
Can I divide it by 4?
Yes, then 4 is a factor
Can I divide it by 5?
No, then 5 is NOT a factor.

Factors of 24: 1,2,3,4,6,8,12,24

Prime Numbers

Prime numbers only have two factors 1 and itself.

For example: 2,3,5,7,11,13,17,19,23...

are prime numbers because their only factors are 1 and itself.

2: 1,2

3: 1,3

5: 1,5

7: 1,7

It is important to identify a prime factor by using the divisibility rules.