Integers review

Whole numbers

Counting numbers

Integers

The number line

Fractions

Equivalent fractions

Adding fractions with the same denominator

Adding fraction with a different denominator

The three types of fractions

Converting mixed fractions to improper fractions

Converting improper fractions to mixed fractions

Multiplying fractions

Multiplying mixed fractions

Reciprocal of a fraction

Dividing fractions

Decimals, percents and fractions on the number line

Prime and composite numbers

Prime factorization

Factor tree

Greatest common factor

Lowest common multiple

Decimal positions

Rounding of decimals

Order of operations (BEDMAS)

Rational numbers

Exponents

Scientific notation

***This review package was made from material that can be found at mathisfun.com***

Whole numbers: Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ...

No Fractions!

Counting numbers: Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero.

So they are 1, 2, 3, 4, 5, ... (and so on).

Integers: Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed!

So, integers can be negative {-1, -2,-3, -4, -5, ...}, positive {1, 2, 3, 4, 5, ...}, or zero {0}.

We can put that all together like this:

Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }

Summary:

The number line:

Writing numbers down on a Number Line makes it easy to tell which numbers are bigger or smaller.

Negative Numbers (−) Positive Numbers (+)

(The line continues left and right forever.)

A number on the left is smaller than a number on the right.

Examples:

5 is smaller than 8

−1 is smaller than 1

−8 is smaller than −5

A number on the right is larger than a number on the left.

Examples:

8 is larger than 5

1 is larger than −1

−5 is larger than −8

Fractions: A fraction is part of a whole. Ex: 1

2

We call the top number, 1, the Numerator, it is the number of parts we have.

We call the bottom number, 2, the Denominator, it is the number of parts the

whole is divided into.

Numerator

Denominator

Equivalent Fractions:

Some fractions may look different, but are really the same, for example:

4/8 = 2/4 = 1/2

(Four-Eighths) Two-Quarters) (One-Half)

=

=

It is usually best to show an answer using the simplest fraction ( 1/2 in this

case ). That is called Simplifying, or Reducing the Fraction

Adding Fractions with the same denominator

It is easy to add fractions with the same denominator (same bottom

number):

1/4 + 1/4 = 2/4 = 1/2

(One-Quarter) (One-Quarter) (Two-Quarters) (One-Half)

+

=

=

Adding Fractions with Different Denominators

But what about when the denominators (the bottom numbers) are not the

same?

We can't add fractions with different denominators:

1

3 +

1

6 = ?

You must find the Least Common Denominator:

1

3 List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...

1

6 List the multiples of 6: 6, 12, 18, 24, ...

Then find the smallest number that is the same

multiples of 3: 3, 6 , 9, 12, 15, 18, 21, ...

multiples of 6: 6 , 12, 18, 24, ...

Three Types of Fractions

There are three types of fraction:

Converting Mixed Fractions to Improper Fractions

To convert a mixed fraction to an improper fraction, follow these steps:

Multiply the whole number part by the fraction's denominator.

Add that to the numerator

Write that result on top of the denominator.

Example: Convert 3 2/5 to an improper fraction.

Multiply the whole number (3) by the fraction's denominator (5):

3 × 5 = 15

Add the fraction's numerator (2) to that:

15 + 2 = 17

Then put that above the denominator, like this:

17

5

Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

Write down the whole number answer

Then write down any remainder above the denominator.

Example: Convert 11/4 to a mixed fraction.

Divide:

11 ÷ 4 = 2 with a remainder of 3

Write down the 2 and then write down the remainder (3) above the

denominator (4), like this

2 3

4

Multiplying fractions

1. Multiply the top numbers (the numerators).

2. Multiply the bottom numbers (the denominators).

3. Simplify the fraction if needed.

Multiplying mixed fractions

Step, by step it is:

Convert both to improper fractions

1 1/2 × 2 1/5 = 3/2 × 11/5

Multiply the fraction (multiply the top numbers, multiply bottom numbers):

3/2 × 11/5 = (3 × 11)/(2 × 5) = 33/10

Convert to a mixed number

33/10 = 3 3/10

The reciprocal of a fraction

To get the reciprocal of a fraction, just turn it upside down. In other words swap over the Numerator and Denominator.

Examples:

Dividing fractions

Example:

Decimals, percents and fractions on the number line

Visit this website for the interactive explanation:

https://www.mathsisfun.com/numbers/decimal-percent-fraction-line.html

Prime and composite numbers:

Prime Number can be divided evenly only by 1 or itself.

And it must be a whole number greater than 1.

Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it

is a composite number).

Prime factorization

"Prime Factorization" is finding which prime numbers multiply together to

make the original number.

Here are two examples:

1) What is the prime factorization of 147 ?

Can we divide 147 evenly by 2?

147 ÷ 2 = 73½

No it can't. The answer should be a whole number, and 73½ is not.

Let's try the next prime number, 3:

147 ÷ 3 = 49

That worked, now we try factoring 49, and find that 7 is the smallest prime

number that works:

49 ÷ 7 = 7

And that is as far as we need to go, because all the factors are prime

numbers.

147 = 3 × 7 × 7

(or 147 = 3 × 72 using exponents)

2) What is the prime factorization of 17 ?

Hang on ... 17 is a Prime Number.

So that is as far as we can go.

17 = 17

Factor Tree

And a "Factor Tree" can help: find any factors of the number, then the

factors of those numbers, etc, until we can't factor any more.

Example: 48

48 = 8 × 6, so we write down "8" and "6" below 48

Now we continue and factor 8 into 4 × 2

Then 4 into 2 × 2

And lastly 6 into 3 × 2

We can't factor any more, so we have found the prime factors.

Which reveals that 48 = 2 × 2 × 2 × 2 × 3

(or 48 = 24 × 3 using exponents)

Greatest Common Factor

The highest number that divides exactly into two or more numbers.

It is the "greatest" thing for simplifying fractions!

Let's start with an Example ...

Greatest Common Factor of 12 and 16

1. Find all the Factors of each number,

2. Circle the Common factors,

3. Choose the Greatest of those

Least (Lowest) Common Multiple

The smallest positive number that is a multiple of two or more numbers.

Least Common Multiple of 3 and 5:

List the Multiples of each number,

The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc

The multiples of 5 are 5, 10, 15, 20, 25, ... etc

Find the first Common (same) value:

The Least Common Multiple of 3 and 5 is 15

( 15 is a common multiple of 3 and 5, and is the smallest, or least, common

multiple )

Decimal positions

Rounding example:

Round 6283.854 to the nearest tenth position = 6283.9

Order of operations:

BEDMAS

Brackets

Exponents

Division/multiplication (Reason why these are with each other is because

you do multiplication or division in the order that you read it from left to

right).

Addition/subtraction (Reason why these are with each other is because

you do addition or substration in the order that you read it from left to right).

Example:

7 + (6 × 52 + 3)

= 7 + (6 x 25 + 3)

= 7 + (150 + 3)

= 7 + 153

= 160

Rational Numbers

A Rational Number is a number that can be written as a simple fraction

(ex: as a ratio).

Most numbers we use in everyday life are Rational Numbers.

Example:

1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

Here are some more examples:

Number As a Fraction Rational?

5 5/1 Yes

1.75 7/4 Yes

.001 1/1000 Yes

-0.1 -1/10 Yes

0.111... 1/9 Yes

√2

(square root of 2) ? NO !

Exponents

The exponent of a number says how many times to

use the number in a multiplication.

In 82 the "2" says to use 8 twice in a multiplication,

so 82 = 8 × 8 = 64

In words: 82 could be called "8 to the power 2" or simply "8 squared"

Exponents are also called Powers or Indices.

Some more examples:

Example:

53 = exponential form

5 × 5 × 5 = expanded form

125 = standard form (answer)

In words: 53 could be called "5 to the power 3" or simply "5 cubed"

Example:

24 = exponential form

2 × 2 × 2 × 2 = expanded form

16 = standard form (answer)

In words: 24 could be called "2 to the power 4"

Negative Exponents

Negative is the opposite of positive, and dividing is the opposite of

multiplying.

A negative exponent means how many times to divide 1 by the number.

Example: 8-1 = 1 ÷ 8 = 0.125

You can have many divides:

Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:

1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

A negative exponent is 1 divided by that number:

8-1 simply means 𝟏

𝟖 , and

8-2 means 𝟏

𝟖 ×𝟖=

𝟏

𝟖𝟐

Scientific notation

Scientific notation is a special way of writing numbers. It makes it easy to

use big and small values.

Let’s look at an example of a large number that you want to write in

scientific notation:

Example: 4205 = 4.205 x 103

Steps:

I. Find the decimal to begin. Here the decimal to begin was behind

the 5 in the number 4205 like so,

4205 = 4205.

II. Move the decimal so that it ends up behind the 1st digit. When it is a

large number, you move the decimal to the left. In this case it was

moved behind the 4 like so,

4205. = 4.205

III. Now you multiply by the Power of 10. Ask yourself the question “How

many places did I move the decimal?” In this case you moved it three

places to the left. This means that the power of 10 is 3, like so

4.205 x 103 = scientific notation

Now to go from scientific notation to standard form, you simply move

the decimal to the right three times (the opposite direction) to make the

original large number

4.205 x 103 = 4205

Let’s look at an example of a small number that you want to write in

scientific notation:

Example: 0.000004205 = 4.205 x 10-6

Steps:

I. Find the decimal to begin.

0.000004205

II. Move the decimal so that it ends up behind the 1st digit. When it is a

small number, you move the decimal to the right. In this case it was

moved behind the 4 like so,

0.000004205 = 4.205

III. Now you multiply by the Power of 10. Ask yourself the question “How

many places did I move the decimal?” In this case you moved it six

places to the right. This means that the power of 10 is -6, like so

4.205 x 10-6 = scientific notation

Now to go from scientific notation to standard form, you simply move

the decimal to the left six times (the opposite direction) to make the original

small number

0000004.205 x 10-6 = 0.000004205