Binary Number System

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

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. 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

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Decimal vs Binary
Here are some equivalent values:

Symmetry
Binary numbers also have a beautiful and elegant pattern:

Here are some larger values: "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.

Example: 10,1
You can do conversions at Binary to Decimal to Hexadecimal Converter.
 * The "10" means 2 in decimal,
 * The ".1" means half,
 * So "10,1" in binary is 2,5 in decimal

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 "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).

Example: What is 11112 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 10012 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 1,12 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 10,112 in Decimal?
"There are 10 kinds of people in the world, those who understand binary numbers, and those who don't."
 * 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