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Integers review
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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
(Four-Eighths)
=
2
/4
Two-Quarters)
=
=
1
/2
(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
1
2
/4
+
/4
=
/4
=
(One-Quarter)
(One-Quarter)
(Two-Quarters)
+
=
=
1
/2
(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
1
6
List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
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
3
2
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
8-2 means
𝟏
=
𝟖 ×𝟖
𝟏
𝟖
, and
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
II.
4205.
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.
III.
=
=
4.205
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
III.
=
4.205
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