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 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

Adding and Subtracting

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,

THEN THE ANSWER IS ZERO.

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

Step 5: Addition:

73 -4

Step 6: Subtraction:

69

Final Answer: 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

Divisibility rules will help you identify the factors of a number.

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.