# data[].attributes.headline
PurpleMath: Mixed Numbers and Improper Fractions

# data[].attributes.subject
PurpleMath: Mixed Numbers and Improper Fractions

# data[].attributes.attachment_link.description
So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.

# data[].attributes.attachment_link.readable_text
Numbers Can be Positive or Negative

This is the Number Line:

Negative Numbers (−)

Positive Numbers (+)

"−" is the negative sign.

"+" is the positive sign

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Play with it!

First, try the sliders below and see what happens when numbers go negative:

numbers/images/number-line-add.js

Balloons and Weights

Let us think about numbers as balloons (positive) and weights (negative):

This basket has balloons and weights tied to it:

The balloons pull up (positive)

And the weights drag down (negative)

Adding a Positive Number

Adding positive numbers is just simple addition.

We can add balloons (we are adding positive value)

the basket gets pulled upwards (positive)

is really saying

Subtracting A Positive Number

Subtracting positive numbers is just simple subtraction.

We can take away balloons (we are subtracting positive value)

the basket gets pulled downwards (negative)

is really saying

Adding A Negative Number

Now let's see what adding and subtracting negative numbers looks like:

We can add weights (we are adding negative values)

the basket gets pulled downwards (negative)

Example: 6 + (−3) = 3

is really saying

"Positive 6 plus Negative 3 equals Positive 3"

We could write it as (+6) + (−3) = (+3)

The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.

So these have the same result:

In other words subtracting a positive is the same as adding a negative.

Subtracting A Negative Number

Lastly, we can take away weights (we are subtracting negative values)

the basket gets pulled upwards (positive)

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Yes indeed! Subtracting a Negative is the same as adding!

Two Negatives make a Positive

What Did We Find?

Adding a positive number is simple addition ...

Adding a Positive is Addition

Positive and Negative Together ...

Example: What is 6 − (+3) ?

6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7) ?

Subtracting a negative ...

Example: What is 14 − (−4) ?

14−(−4) = 14 + 4 = 18

The Rules:

It can all be put into two rules:

Rule

Example

+(+)

Two like signs become a positive sign

3+(+2) = 3 + 2 = 5

−(−)

6−(−3) = 6 + 3 = 9

+(−)

Two unlike signs become a negative sign

7+(−2) = 7 − 2 = 5

−(+)

8−(+2) = 8 − 2 = 6

They are "like signs" when they are like each other (in other words: the same).

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign

Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign.

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Starting Negative

What if we start with a negative number?

Using The Number Line can help:

Example: What is −3+(+2) ?

+(+) are like signs, so they become a positive sign.

Start at −3 on the number line,
move forward 2 and you end up at −1

Example: What is −3+(−2) ?

+(−) are unlike signs, so they become a negative sign.

Start at −3 on the number line,
move back 2 and you end up at −5

Now Play With It!

Try playing Casey Runner, you need to know the rules of positive and negative to succeed!

A Common Sense Explanation

And there is a "common sense" explanation:

If I say "Eat!" I am encouraging you to eat (positive)

If I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you are done!

Another Common Sense Explanation

A Bank Example

Example: Last year the bank subtracted $10 from your account by mistake, and they want to fix it.

So the bank must take away a negative $10.

Let's say your current balance is $80, so you will then have:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

A Long Example You Might Like

Ally's Points

Ally can be naughty or nice. So Ally's parents have said
"If you are nice we will add 3 points (+3).
If you are naughty, we take away 3 points (−3).
When you reach 30 Points you get a toy."

Then Dad confesses he spilt the milk and writes "undo".

How do we "undo" a minus 3?
We add 3 back again!

So when we subtract a negative, we gain points
(i.e. the same as adding points).

See: both " 15 − (+3)" and " 15 + (−3)" result in 12.

So:

It doesn't matter if you subtract positive points
or add negative points,
you still end up losing points.

Try These Exercises ...

Now try This Worksheet, and see how you go.

And also try these questions:

11715, 11716, 11717, 11718, 11719, 11720, 11721, 3445, 3446

# data[].attributes.attachment_link.title
Adding and Subtracting Positive and Negative Numbers

# data[].attributes.attachment_link.description
This is the Number Line: Negative Numbers (−) Positive Numbers (+) "−" is the negative sign. "+" is the positive sign If a number has no sign it usually means that it is a positive number.

# data[].attributes.attachment_link.description
Below is the process written out in full. You will often see other versions, which are generally just a shortened version of the process below. You can also see this done in Long Division Animation.

# data[].attributes.attachment_link.readable_text
Long Multiplication is a special method for multiplying larger numbers.

It is a way to multiply numbers larger than 10 that only needs your knowledge of the ten times Multiplication Table.

Let us say we want to multiply

612 × 24

First we multiply 612 × 4 (=2,448),

then we multiply 612 × 20 (=12,240),

and last we add them together (2,448 + 12,240 = 14,688).

But we can do better!

When we multiply 612 × 20 we only need to multiply and place the result one column over (so it is the same as multiplying by 20).

We just have to be careful about lining up the columns. Here is how to do it (press the play button):

We could also put a zero where the space is, like this:

More Than Two Digits

The same idea works when multiplying by more than two digit numbers. We just have to move over more spaces:

Have a try yourself with these Long Multiplication Worksheets

# data[].attributes.attachment_link.description
Long Multiplication is a special method for multiplying larger numbers. It is a way to multiply numbers larger than 10 that only needs your knowledge of the ten times Multiplication Table. Let us say we want to multiply 612 × 24 First we multiply 612 × 4 (=2,448), then we multiply 612 × 20 (=12,240), and last we add them together (2,448 + 12,240 = 14,688).

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.attachment_link.readable_text
A fraction is a part of a whole

You can learn about:

Fractions

Introduction to Fractions

Decimals, Fractions and Percentages

Interactive Fractions

Fractions on a pizza and the number line

Decimals, Fractions and Percentages on the Number Line

Words to Pizza

Words to Number Line

Fraction to Pizza

Fraction to Number Line

Fractions on a Clock Face

The three different types of fractions

Proper Fractions

Improper Fractions

Mixed Fractions

How to simplify a fraction, and how the "Greatest Common Factor" can help

Equivalent Fractions

Fraction Number Line

Simplifying Fractions

Greatest Common Factor

Greatest Common Factor Tool

How to compare fractions, and how the "Least Common Multiple" can help

Comparing Fractions

Least Common Multiple

Least Common Multiple Tool

Least Common Denominator

How to add, subtract, multiply and divide fractions

♫ "Multiplying fractions no big problem,
Top times top over bottom times bottom,

"Dividing fractions, easy as pie,
Flip the second fraction, then multiply,

"If adding or subtracting is your aim,
The bottom numbers must be the same!

"Change the bottom using multiply or divide,
But the same to the top must be applied,

"And don't forget to simplify,
Before it's time to say goodbye"  ♫

Multiplying Fractions

Dividing Fractions

Adding Fractions Adding Fractions Illustrated

Subtracting Fractions Subtracting Fractions Illustrated

Dividing Fractions by Whole Numbers

Adding Mixed Fractions

Multiplying Mixed Fractions

How to convert fractions to decimals and vice versa

Convert Decimals to Fractions and Decimal to Fraction Converter

Converting Fractions to Decimals

How to convert fractions to percentages and vice versa

Convert Percentages to Fractions

Converting Fractions to Percentages

Reciprocal fractions

Reciprocal Fractions