Binary Number System

A Binary Number is made up of only **0**s and **1**s.

 Example of a Binary Number

There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!

A "**bit**" is a single **b**inary dig**it**. The number above has 6 bits.

Binary numbers have many uses in mathematics and beyond.

In fact the digital world uses binary digits.

## How do we Count using Binary?

It is just like counting in decimal except we reach 10 much sooner.

 1
 ???

The same thing is done in binary ...

And that is what we do in binary ...

See how it is done in this little demonstration (press play button):

Base:

Hexadecimal

Decimal

Binary

0

128 = **27**

0

64 = **26**

0

32 = **25**

0

16 = **24**

0

8 = **23**

0

4 = **22**

0

2 = **21**

0

1 = **20**

© 2018 MathsIsFun.com v0,82

"Binary is as easy as 1, 10, 11."

Now see how to use Binary to count past 1.000 on your fingers:

Activity: Binary Fingers

## Position

In the Decimal System there are Ones, Tens, Hundreds, etc

In **Binary** there are Ones, Twos, Fours, etc, like this:

This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8)
= **13,625 in Decimal**

Numbers can be placed to the left or right of the point, to show values greater than one and less than one.

gets **2 times smaller** (half as big).

### Example: 10,1

* The "10" means 2 in decimal,
* The ".1" means half,
* So "10,1" in binary is 2,5 in decimal

You can do conversions at Binary to Decimal to Hexadecimal Converter.

## Words

The word **binary** comes from "Bi-" meaning two. We see "bi-" in words such as "bicycle" (two wheels) or "binocular" (two eyes).

A single binary digit (like "0" or "1") is called a "bit".

For example **11010** is five bits long.

The word**bit** is made up from the words "**b**inary dig**it**"

## How to Show that a Number is Binary

To show that a number is a *binary* number, follow it with a little 2 like this: **1012**

This way people won't think it is the decimal number "101" (one hundred and one).

## Examples

### Example: What is 1111**2** in Decimal?

* The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)
* The next "1" is in the "2×2" position, so that means 1×2×2 (=4)
* The next "1" is in the "2" position, so that means 1×2 (=2)
* The last "1" is in the ones position, so that means 1
* Answer: 1111 = 8+4+2+1 = 15 in Decimal

### Example: What is 1001**2** in Decimal?

* The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)
* The "0" is in the "2×2" position, so that means 0×2×2 (=0)
* The next "0" is in the "2" position, so that means 0×2 (=0)
* The last "1" is in the ones position, so that means 1
* Answer: 1001 = 8+0+0+1 = 9 in Decimal

### Example: What is 1,1**2** in Decimal?

* The "1" on the left side is in the ones position, so that means 1.
* The 1 on the right side is in the "halves" position, so that means 1×(1/2)
* So, 1,1 is "1 and 1 half" = 1,5 in Decimal

### Example: What is 10,11**2** in Decimal?

* The "1" is in the "2" position, so that means 1×2 (=2)
* The "0" is in the ones position, so that means 0
* The "1" on the right of the point is in the "halves" position, so that means 1×(1/2)
* The last "1" on the right side is in the "quarters" position, so that means 1×(1/4)
* So, 10,11 is 2+0+1/2+1/4 = 2,75 in Decimal

"There are 10 kinds of people in the world,
those who understand binary numbers, and those who don't."