Binary Number System

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

110100
Example of a Binary Number

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

A "bit" is a single binary digit. The number above has 6 bits.

Binary numbers have many uses in mathematics and beyond.

digital world
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.

Binary  
0 We start at 0
1 Then 1
??? But then there is no symbol for 2 ... what do we do?

 

Well how do we count in Decimal?
 0 Start at 0
 ... Count 1,2,3,4,5,6,7,8, and then...
 9 This is the last digit in Decimal
 10 So we start back at 0 again, but add 1 on the left

The same thing is done in binary ...

 Binary  
 0 Start at 0
1 Then 1
••10 Now start back at 0 again, but add 1 on the left
•••11 1 more
••••??? But NOW what ... ?

 

What happens in Decimal?
 99 When we run out of digits, we ...
 100 ... start back at 0 again, but add 1 on the left

And that is what we do in binary ...

 Binary  
 0 Start at 0
1 Then 1
••10 Start back at 0 again, but add 1 on the left
•••11  
••••100 start back at 0 again, and add one to the number on the left...
... but that number is already at 1 so it also goes back to 0 ...
... and 1 is added to the next position on the left
•••••101  
••••••110  
•••••••111  
••••••••1000 Start back at 0 again (for all 3 digits),
add 1 on the left
•••••••••1001 And so on!

 

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

Decimal vs Binary

Here are some equivalent values:

Decimal:0123456789101112131415
Binary:01101110010111011110001001101010111100110111101111

Symmetry

yin yang

Binary numbers also have a beautiful and elegant pattern:

binary symmetry


Here are some larger values:

Decimal:2025304050100200500
Binary:101001100111110101000110010110010011001000111110100

"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:

binary number 1101.101

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.

10.1
left arrowdot blueThe number to the left of the point is a whole number (such as 10)
  
As we move further left, every number place
gets 2 times bigger.
  
dot blueright arrowThe first digit on the right means halves (1/2).
  
 As we move further right, every number place
gets 2 times smaller (half as big).

Example: 10.1

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).

speechWhen you say a binary number, pronounce each digit (example, the binary number "101" is spoken as "one zero one", or sometimes "one-oh-one"). This way people don't get confused with the decimal number.

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 "binary digit"

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 11112 in Decimal?

Example: What is 10012 in Decimal?

Example: What is 1.12 in Decimal?

Example: What is 10.112 in Decimal?

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