Number System: Long Division

Long Division

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.

Let's see how it is done with:

• the number to be divided into is known as the dividend
• The number which divides the other number is known as the divisor

And here we go:

	4 ÷ 25 = 0 remainder 4	The first digit of the dividend (4) is divided by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 0 = 0	The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into.
	4 – 0 = 4	Now we subtract the bottom number from the top number.
		Bring down the next digit of the dividend.
	42 ÷ 25 = 1 remainder 17	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 1 = 25	The answer from the above operation is multiplied by the divisor. The result is placed under the last number divided into.
	42 – 25 = 17	Now we subtract the bottom number from the top number.
		Bring down the next digit of the dividend.
	175 ÷ 25 = 7 remainder 0	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 7 = 175	The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into.
	175 – 175 = 0	Now we subtract the bottom number from the top number.
		There are no more digits to bring down. The answer must be 17

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Number System: Dividing Decimals

Dividing Decimals

Quick method: use Long Division without the decimal point,
then re-insert the decimal point in the answer.

Dividing a Decimal Number by a Whole Number

To divide a decimal number by a whole number:

• Use Division or Long Division (ignoring the decimal point)
• Then put the decimal point in the same spot as the dividend (the number being divided)
  

Example: Divide 9.1 by 7

Ignore the decimal point and use Long Division:

13 7 )91    9    7    21    21     0

Put the decimal point in the answer directly above the decimal point in the dividend:

1.3 7 )9.1

The answer is 1.3

Dividing by a Decimal Number

But what if you want to divide by a Decimal Number?
The trick is to convert the number you are dividing by to a whole number first, by shifting the decimal point of both numbers to the right:

Now you are dividing by a whole number, and can continue as normal.
It is safe to do this if you remember to shift the decimal point of both numbers the same number of places.

Example: Divide 6.4 by 0.4

You are not dividing by a whole number, so you need to move the decimal point so that you are dividing by a whole number:

move 1
6.4		64
0.4		4
move 1

6.4/0.4 is exactly the same as 64/4,
as you moved the decimal point of both numbers.

And the answer is:

64 / 4 = 16

You can see for yourself how many 0.4s make 6.4:

Example: Divide 5.39 by 1.1

Move the decimal point so that you are dividing by a whole number:

move 1
5.39		53.9
1.1		11
move 1

You are now dividing by a whole number, so you can proceed:
Ignore the decimal point and use Long Division:

049 11 )539     5     0     53     44      99      99       0

Put the decimal point in the answer directly above the decimal point in the dividend:

04.9 11 )53.9

The answer is 4.9

Animations

Have a look at these Decimal Division Animations for further help.

Lastly ...

As a final check you can put your "common sense" hat on and think "is that the right size?", because you don't want to pay ten times too much for anything, nor do you want to get only one-tenth of what you need!

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Number System: Multiplying Decimals

Multiplying Decimals

Multiply without the decimal point, then re-insert it in the correct spot!

How to Multiply Decimals

Just follow these steps:

• Multiply normally, ignoring the decimal points.
• Then put the decimal point in the answer - it will have as many decimal places as the two original numbers combined.

In other words, just count up how many numbers are after the decimal point in both numbers you are multiplying, then the answer should have that many numbers after its decimal point.

Example: Multiply 0.03 by 1.1

start with:		0.03 × 1.1
multiply without decimal points:		3 × 11 = 33
0.03 has 2 decimal places, and 1.1 has 1 decimal place, so the answer has 3 decimal places:		0.033

See the Multiplying Decimals Animations

How Does It Work?

Because when you multiply without the decimal point, you are really shifting the decimal point to the right to get it out of the way:

Original:		1 Move:		2 Moves:		3 Moves:
0.03 × 1.1		0.3 × 1.1		3. × 1.1		3. × 11.

Then we do the (now easy) multiplication:

3. × 11. = 33.

But remember, we did 3 Moves of the decimal point, so we need to undo that:

3 Moves:		2 Moves:		1 Move:		Correct
33.		3.3		0.33		0.033

Here are some more examples:

Example: Multiply 0.25 by 0.2

	start with:		0.25 × 0.2
	multiply without decimal points:		25 × 2 = 50
	0.25 has 2 decimal places, and 0.2 has 1 decimal place, so the answer has 3 decimal places:		0.050

Example: Multiply 102 by 0.22

	start with:		102 × 0.22
	multiply without decimal points:		102 × 22 = 2,244
	102 has 0 decimal places, and 0.22 has 2 decimal place, so the answer has 2 decimal places:		22.44

Common Sense

As a final check you can put your "common sense" hat on and think "is that the right size?", because you don't want to pay ten times too much for anything, nor get only one-tenth of what you need!
And that's all.

Just remember: the answer should have the same number of decimal places as are in both the numbers you are multiplying.

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Number System: Subtracting Decimals

Subtracting Decimals

Subtracting decimals is easy if you keep your work neat

To subtract decimals, follow these steps:

• Write down the two numbers, one under the other, with the decimal points lined up.
• Add zeros so the numbers have the same length
• Then subtract normally, remembering to put the decimal point in the answer

Example: Subtract 0.03 from 1.1

Line the decimals up:			1.1
		-	0.03
"Pad" with zeros:			1.10
		-	0.03
Subtract:			1.10
		-	0.03
			1.07

Answer: 1.07

So, that was just like 110 - 3 = 107, but with the decimal point in a different position

Example: Calculate 7.005-0.55

Line the decimals up:			7.005
		-	0.55
"Pad" with zeros:			7.005
		-	0.550
Subtract:			7.005
		-	0.550
			6.455

Answer: 6.455

And that was just like 7,005 - 550 = 6,455

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Number System: Adding In Columns

Addition in Columns

You can do addition by writing one number below the other

and then add one column at a time, like this:

(Practice on this Simple Column Addition Worksheets)

And it works for bigger numbers, too:

But sometimes two-digit numbers can get in the way like this:

THIS is how it should be done:

Did you see how we split the 12 into 1 Ten and 2 Units? We then sent the 1 Ten to the top of the Tens column, so we will remember to add it when we do the Tens. And there is a nice clear space for the Tens answer below.

Here's another example (done a little faster):

Now practice on this Column Addition (with Carrying) Worksheets

Above or Below? Your teacher may want you to put the "1" below like in this example. Above or below does not really matter, you just need a good place to put the "1" so you don't forget it!

What About Bigger Numbers?

The same method works with numbers in the hundreds, thousands and so on.
Like this:

View Larger

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Number System: Adding and Subtracting Decimals

Adding and Subtracting Decimals

Adding decimals is easy if you keep your work neat

To add decimals, follow these steps:

• Write down the numbers, one under the other, with the decimal points lined up
• Put in zeros so the numbers have the same length
• Then add using column addition, remembering to put the decimal point in the answer

Example: Add 1.452 to 1.3

Line the decimals up:			1.452
		+	1.3
"Pad" with zeros:			1.452
		+	1.300
Add:			1.452
		+	1.300
			2.752

Example: Add 3.25, 0.075 and 5

Line the decimals up:			3.25
			0.075
		+	5.
"Pad" with zeros:			3.250
			0.075
		+	5.000
Add:			3.250
			0.075
		+	5.000
			8.325

That's all there is to it - just remember to line up the decimals, then add normally.

Subtracting

To subtract, follow the same method: line up the decimals, then subtract.

Example: What is 7.368 − 1.15 ?

Line the decimals up:			7.368
		−	1.15
"Pad" with zeros:			7.368
		−	1.150
Subtract:			7.368
		−	1.150
			6.218

To check we can add the answer to the number subtracted:

Example (continued): Check by adding 6.218 to 1.15

Line the decimals up:			6.218
		+	1.15
"Pad" with zeros:			6.218
		+	1.150
Add:			6.218
		+	1.150
			7.368

It matches the number we started with, so it checks out.

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Number System: Factorial Function

Factorial !

Example: 4! is shorthand for 4 x 3 x 2 x 1

The factorial function (symbol: !) means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 1! = 1

4! is usually pronounced "4 factorial", but some people even say "4 shriek" or "4 bang"

Calculating From the Previous Value

You can easily calculate a factorial from the previous one:

n	n!
1	1	1	1
2	2 × 1	= 2 × 1!	= 2
3	3 × 2 × 1	= 3 × 2!	= 6
4	4 × 3 × 2 × 1	= 4 × 3!	= 24
5	5 × 4 × 3 × 2 × 1	= 5 × 4!	= 120
6	etc	etc

Example: What is 10! if you know that 9!=362,880 ?

10! = 10 × 9!
10! = 10 × 362,880 = 3,628,800

So the rule is:

n! = n × (n-1)!

Which just says "the factorial of any number is that number times the factorial of (1 smaller than that number)", hence 10! = 10 × 9!, or even 125! = 125 × 124!

What About "0!"

Zero Factorial is interesting ... it is generally agreed that 0! = 1.
It may seem funny that multiplying no numbers together gets you 1, but it helps simplify a lot of equations.

Where is Factorial Used?

Factorials are used in many areas of mathematics, but particularly in Combinations and Permutations

Example: What is 7! / 4!

Let us write them out in full:

7 × 6 × 5 × 4 × 3 × 2 × 1		= 7 × 6 × 5 = 210
4 × 3 × 2 × 1

That was neat. The 4 × 3 × 2 × 1 "cancelled out", leaving only 7 × 6 × 5

A Small List

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800
11	39,916,800
12	479,001,600
13	6,227,020,800
14	87,178,291,200
15	1,307,674,368,000
16	20,922,789,888,000
17	355,687,428,096,000
18	6,402,373,705,728,000
19	121,645,100,408,832,000
20	2,432,902,008,176,640,000
21	51,090,942,171,709,440,000
22	1,124,000,727,777,607,680,000
23	25,852,016,738,884,976,640,000
24	620,448,401,733,239,439,360,000
25	15,511,210,043,330,985,984,000,000

As you can see, it gets big quickly!
If you need more, try our Full Precision Calculator.

Some Bigger Values

70! is approximately 1.1978571669969891796072783721 x 10¹⁰⁰, which is just larger than a Googol (the digit 1 followed by one hundred zeros).
100! is approximately 9.3326215443944152681699238856 x 10¹⁵⁷
200! is approximately 7.8865786736479050355236321393 x 10³⁷⁴

... Advanced Topic Follows ...

What About Decimals?

Can you have factorials for numbers like 0.5 or -3.217?
Yes you can! But you need to get into a subject called the "Gamma Function", which is beyond this simple page.

Half Factorial

But I can tell you the factorial of half (½) is half of the square root of pi = (½)√π, and so some "half-integer" factorials are:

n	n!
(-½)!	√π
(½)!	(½)√π
(3/2)!	(3/4)√π
(5/2)!	(15/8)√π

It still follows the rule that "the factorial of any number is that number times the factorial of (1 smaller than that number)", because

(3/2)! = (3/2) × (1/2)!
(5/2)! = (5/2) × (3/2)!

Can you figure out what (7/2)! is?

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Number System: Dividing by Zero

Dividing by Zero

Don't divide by zero or this could happen! Just kidding.

In truth:

Dividing by Zero is undefined.

Dividing

To see why, let us look at what is meant by "division":

Division is splitting into equal parts or groups.
It is the result of "fair sharing".

Example: there are 12 chocolates, and 3 friends want to share them, how do they divide the chocolates?

12 Chocolates		12 Chocolates Divided by 3

So they get 4 each: 12/3 = 4

Dividing by Zero

Now, let us try dividing the 12 chocolates among zero people, how much does each person get?
Does that question even make sense? No, of course it doesn't.
You can't share among zero people, and you can't divide by 0.

Another Good Reason

After dividing, can you multiply to get back again?

But multiplying by 0 gives 0, so that won't work.
Once again, dividing by zero gives us difficulties!

 

Imagine You Could Divide by Zero

Okay, let us imagine you can divide by zero, and see what happens.
That means that things like ¹/₀ and ⁰/₀ would behave like normal numbers.

Try Multiplying By Zero

So let us try using our new "numbers".
For example, we know that zero times any number is zero:

Example: 0×1 = 0, 0×2 = 0, etc

So that should also be true for ¹/₀:

0 × (¹/₀) = 0

But we could also rearrange it a little like this:

0 × (¹/₀) = (⁰/₀) × 1 = 1

(Careful! I am not saying this is correct! , we are assuming that you can divide by zero, so 0/0 should work the same as 5/5, which is 1).
Arrggh! If you multiply ¹/₀ by zero you could get 0 or 1.
In fact you can't have both possibilites, so we cannot define ¹/₀ to be a number.
So it is undefined.

So what is 0/0 ?

0/0 is like asking "how many 0s in 0?"
Are there no zeros in zero at all? Or perhaps there is exactly one zero in zero? Or many zeros?
So 0/0 is indeterminate (it could be any value).

In conclusion:

When you try to divide by zero, things stop making sense

That is all.

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Number System: Even and Odd numbers

Even and Odd Numbers

Even Numbers

Any integer that can be divided exactly by 2.
The last digit will be 0, 2, 4, 6 or 8

Example: -24, 0, 6 and 38 are all even numbers

Odd Numbers

If it is not an even number, it is called an odd number.
The last digit will be 1, 3, 5, 7 or 9

Example: -3, 1, 7 and 35 are all odd numbers

Adding and Subtracting

When you add (or subtract) odd or even numbers the results are always:

Operation	Result	Example (red is odd, blue is even)
Even + Even	Even	2 + 4 = 6
Even + Odd	Odd	6 + 3 = 9
Odd + Even	Odd	5 + 12 = 17
Odd + Odd	Even	3 + 5 = 8

(The same thing happens if you subtract instead of adding)

Multiplying

When you multiply odd or even numbers the results are always:

Operation	Result	Example (red is odd, blue is even)
Even × Even	Even	4 × 8 = 32
Even × Odd	Even	4 × 7 = 28
Odd × Even	Even	5 × 8 = 40
Odd × Odd	Odd	5 × 7 = 35

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Number System: Number Line

Number Line

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

The Number Line

Negative Numbers (-)	Positive Numbers (+)
(The line continues left and right forever.)

Numbers on the left are smaller than numbers on the right. Examples: 5 is smaller than 8 -1 is smaller than 1 -8 is smaller than -5

Try this interactive number line (click to mark):

View Larger

You can also try the zoomable number line.

An Example

Example: John owes $3, Virginia owes $5 but Alex doesn't owe anything, in fact he has $3 in his pocket. Place these people on the number line to find who is poorest and who is richest.

Having money in your pocket is positive. But owing money is negative. So John has "-3", Virginia "-5" and Alex "+3"
Now it is easy to see that Virginia is poorer than John (-5 is less than -3) and John is poorer than Alex (-3 is smaller than 3), and Alex is, of course, the richest!

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Number System: Zero

Zero

Zero shows that there is no amount.

Example: 6 - 6 = 0 (the difference between six and six is zero)

It is also used as a "placeholder" so that you can write a numeral properly.

Example: 502 (five hundred and two) could be mistaken for 52 (fifty two) without the zero in the tens place.

Zero is a very special number ...

It is halfway between -1 and 1 on the Number Line:

Zero is neither negative nor positive, but it is an even number!

The Idea

The idea of zero, though natural to us now, was not natural to early humans ... if there is nothing to count, how can you count it?

Example: you can count dogs, but you can't count an empty space:

Two Dogs		Zero Dogs? Zero Cats?

An empty patch of grass is just an empty patch of grass!

Zero as a Placeholder

But about 3,000 years ago, when people started writing bigger numbers like "42" they had a problem: how to tell the difference between "4" and "40". Without the zero they look the same!
So zero is now used as a "placeholder": it shows "there is no number at this place"

502	Which means 5 hundreds, no tens, and 2 units

The Value of Zero

Then people started thinking of zero as an actual number.

Example:

"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"

Additive Identity

And zero has a special property: when you add it to a number you get that number back, unchanged

Example:

7 + 0 = 7

Adding 0 to 7 gives the answer 7
Also 0 + 7 = 7

This makes it the Additive Identity, which is just a special way of saying "add 0 and you get the identical number you started with".

Special Properties

Here are some of zero's properties:

Property	Example
a + 0 = a	4 + 0 = 4
a − 0 = a	4 − 0 = 4
a × 0 = 0	6 × 0 = 0
0 / a = 0	0/3 = 0
a / 0 = undefined (dividing by zero is undefined)	7/0 = undefined
0a = 0 (a is positive)	04 = 0
00 = indeterminate	00 = indeterminate
0a = undefined (a is negative)	0-2 = undefined
0! = 1 ("!" is the factorial function)	0! = 1

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Number System: Whole Numbers and Integers

Whole Numbers and Integers

Whole Numbers

Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on)

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).

Natural Numbers

"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.

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, ... }

Example, these are all integers:

-16, -3, 0, 1, 198

(But numbers like ½, 1.1 and 3.5 are not integers)

 

Confusing

Just to be confusing, some people say that whole numbers can also be negative, so that would make them exactly the same as integers. And sometimes people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing!

My Standard

I must admit that sometimes I say "negative whole number", but usually I stick to:

Name	Numbers	Examples
Whole Numbers	{ 0, 1, 2, 3, 4, 5, … }	0, 27, 398, 2345
Counting Numbers	{ 1, 2, 3, 4, 5, … }	1, 18, 27, 2061
Integers	{ ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … }	-15, 0, 27, 1102

But nobody disagrees on the definition of an integer, so when in doubt say "integer", and if you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent. Like this (note: zero is neither positive nor negative):

• Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
• Negative Integers = { ..., -5, -4, -3, -2, -1 }
  
• Positive Integers = { 1, 2, 3, 4, 5, ... }
  
• Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?)

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Number System: Ordering Decimals Number

Ordering Decimals

"Could I have a 3.65 and an 0.8, please ... ?"
NO, not THAT type of ordering. I mean putting them in order ...

Ordering decimals can be tricky. Because often we look at 0.42 and 0.402 and say that 0.402 must be bigger because there are more digits. But no!
If you follow the following method you will see which decimals are bigger. 

• Set up a table with the decimal point in the same place for each number.
• Put in each number.
• Fill in the empty squares with zeros.
• Compare using the first column on the left
• If the digits are equal move to the next column to the right until one number wins.

	If you want ascending order you always pick the smallest first
	If you want decending order you always pick the largest first

Example: Put the following decimals in ascending order:

1.506, 1.56, 0.8

In a table they look like this:

Units	Decimal Point	Tenths	Hundredths	Thousandths
1	.	5	0	6
1	.	5	6
0	.	8

Fill in the empty squares with zeros:

Units	Decimal Point	Tenths	Hundredths	Thousandths
1	.	5	0	6
1	.	5	6	0
0	.	8	0	0

Compare using the first column (Units)

Two of them are "1"s and the other is a "0". Ascending order needs smallest first, and so "0" is the winner:

Answer so far: 0.8

Now we can remove 0.8 from the list:

Units	Decimal Point	Tenths	Hundredths	Thousandths
1	.	5	0	6
1	.	5	6	0
-	.	-	-	-

Compare the Tenths

Now there are two numbers with the same "Tenths" value of 5, so move along to the "Hundredths" for the tie-breaker

Compare the Hundredths

One of those has a 6 in the hundredths, and the other has a 0, so the 6 wins. So 1.506 is less than 1.56:

Answer so far: 0.8, 1.506

Remove 0.506 from the list:

Units	Decimal Point	Tenths	Hundredths	Thousandths
-	.	-	-	-
1	.	5	6	0
-	.	-	-	-

Only one number left, it must be the largest:

Answer: 0.8, 1.506, 1.56

Done!

Example: Put the following decimals in DESCENDING order:

0.402, 0.42, 0.375, 1.2, 0.85

In a table they look like this:

Units	Decimal Point	Tenths	Hundredths	Thousandths
0	.	4	0	2
0	.	4	2
0	.	3	7	5
1	.	2
0	.	8	5

And we want to go from highest to lowest (descending).

Fill in the empty squares with zeros:

Units	Decimal Point	Tenths	Hundredths	Thousandths
0	.	4	0	2
0	.	4	2	0
0	.	3	7	5
1	.	2	0	0
0	.	8	5	0

Compare using the first column (Units):

There is a 1, all the rest are 0. Descending order needs largest first, so 1.2 must be the highest. (Write it down in your answer and cross it off the table).

Answer so far: 1.2

Units	Decimal Point	Tenths	Hundredths	Thousandths
0	.	4	0	2
0	.	4	2	0
0	.	3	7	5
-	-	-	-	-
0	.	8	5	0

Compare the Tenths.

The 8 is highest, so 0.85 is next in value.

Answer so far: 1.2, 0.85

Units	Decimal Point	Tenths	Hundredths	Thousandths
0	.	4	0	2
0	.	4	2	0
0	.	3	7	5
-	-	-	-	-
-	-	-	-	-

Now there are two numbers with the same "Tenths" value of 4, so move along to the "Hundredths" for the tie-breaker
One number has a 2 in the hundredths, and the other has a 0, so the 2 wins. So 0.42 is bigger than 0.402:

Answer so far: 1.2, 0.85, 0.42, 0.402

Units	Decimal Point	Tenths	Hundredths	Thousandths
-	-	-	-	-
-	-	-	-	-
0	.	3	7	5
-	-	-	-	-
-	-	-	-	-

Only 0.375 left, so the answer is:

Answer: 1.2, 0.85, 0.42, 0.402, 0.375

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Number System: Ordering Numbers

Ordering Numbers

"Waiter, I would like a 7 and a 3, please..."
NO, not THAT type of ordering. I mean putting them in order ...

To put numbers in order, place them from lowest (first) to highest (last). This is called "Ascending Order" (think of ascending a mountain)

Example: Place 17, 5, 9 and 8 in ascending order. Answer: 5, 8, 9, 17

Sometimes you want the numbers to go the other way, from highest down to lowest, this is called "Descending Order" (think of a "steep descent")

Example: Place 17, 5, 9 and 8 in descending order. Answer: 17, 9, 8, 5

Practice by neatening up your friends. Measure their heights, then place them in ascending order of height. Try it again, but use their weights.

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Number System: Convert from Decimals to Fractions

Convert Decimals to Fractions

(Multiply top and bottom by 10 until you get a whole number, then simplify)

To convert a Decimal to a Fraction follow these steps:

Step 1: Write down the decimal divided by 1, like this: decimal/1

Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)

Step 3: Simplify (or reduce) the fraction

Example: Express 0.75 as a fraction

Step 1: Write down 0.75 divided by 1:

0.75

1

Step 2: Multiply both top and bottom by 100 (there were 2 digits after the decimal point so that is 10×10=100):

× 100
0.75	=	75
1		100
× 100

(Do you see how it turns the top number
into a whole number?)

Step 3: Simplify the fraction (this took me two steps):

	÷5		÷ 5
75	=	15	=	3
100		20		4
	÷5		÷ 5

Answer = ³/₄

Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction !

Example: Express 0.625 as a fraction

Step 1: write down:

0.625

1

Step 2: multiply both top and bottom by 1,000 (there were 3 digits after the decimal point so that is 10×10×10=1,000)

625

1,000

Step 3: Simplify the fraction (it took me two steps here):

	÷ 25		÷ 5
625	=	25	=	5
1,000		40		8
	÷ 25		÷ 5

Answer = ⁵/₈

Example: Express 0.333 as a fraction

Step 1: Write down:

0.333

1

Step 2: Multiply both top and bottom by 1,000 (there were 3 digits after the decimal point so that is 10×10×10=1,000)

333

1,000

Step 3: Simplify Fraction:

Can't get any simpler!

Answer = ³³³/_1,000

But a Special Note:

If you really meant 0.333... (in other words 3s repeating forever which is called 3 recurring) then we need to follow a special argument. In this case we would write down:

0.333...

1

Then MULTIPLY both top and bottom by 3:

× 3
0.333...	=	0.999...
1		3
× 3

And 0.999... = 1 (Does it? - see the 9 Recurring discussion for more if you are interested), so:

Answer = 1/3

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