Binary Number System

A binary number is made up of only 0s and 1s.

110100
Example of a binary Number

There's no 2, 3, 4, 5, 6, 7, 8 or 9 in binary!

Binary numbers have many uses in mathematics and beyond.

Circuit board with glowing binary numbers 0 and 1
In fact the digital world uses binary digits.

How do we Count using Binary?

It is like counting in decimal except we run out of digits (only 0 and 1) much sooner.

Binary    
0   We start at 0
1   Then 1
???   But then there's 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's 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):

numbers/images/number-odometer.js?mode=2
Binary "10" does not mean the same as "ten" in decimal. It is saying we have 1 lot of the base (2) and 0 ones, or 1 × 2 + 0 = 2

Decimal vs Binary

Here are some equivalent values:

Decimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111

Symmetry

Yin-yang symbol of black and white duality

Binary numbers also have a beautiful and elegant pattern:

0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1

Here are some larger values:

Decimal: 20 25 30 40 50 100 200 500
Binary: 10100 11001 11110 101000 110010 1100100 11001000 111110100

"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, and so on

In binary there are Ones, Twos, Fours, and so on, like this:

Place Value: 8 4 2 1
Example: 1 1 0 1

This is 1×8 + 1×4 + 0×2 + 1
= 13 in Decimal

So 1101 in Binary = 13 in Decimal

We can go beyond the point:

Place value chart for binary 1101.101 with values from 8 to one eighth

This is 1×8 + 1×4 + 0×2 + 1 + 1×(12) + 0×(14) + 1×(18)
= 13.625 in Decimal

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

10.1
Left arrow pointing toward whole number place valuesdot blue The number left of the point is a whole number (in this example 10)
   
As we move further left, every number place gets 2 times bigger.
   
dot blueRight arrow pointing toward fractional place values The first digit on the right means halves (1/2)
   
  As we move further right, every number place gets 2 times smaller (half as big)
The point is just a point (like a decimal point). What changes is the place values: in binary they are powers of 2.

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.

Doubling

Doubling each time gets us:

1 → 2 → 4 → 8 → 16 → 32 → 64 → 128 → 256 → 512 → 1024 → 2048 → 4096 → 8192 → 16,384 → 32,768 → 65,536

Which looks like this in the binary place value table:

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
65,536 32,768 16,384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1

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"

And:

When reading a binary number aloud, say each digit.

Examples:

  • Binary "10" is read as "one zero" (not "ten")
  • Binary "101" is read as "one zero one"

This stops people thinking it is a decimal number.

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's 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: 11112 = 8+4+2+1 = 15 in Decimal

Example: What's 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: 10012 = 8+0+0+1 = 9 in Decimal

Example: What's 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, 1.12 is "1 and 1 half" = 1.5 in Decimal

Example: What's 10.112 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, 10.112 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."

How to Convert Decimal to Binary

To turn a decimal number into binary, find the largest power of 2 that fits into it, subtract it, and keep going until nothing is left:

Example: Convert 13 to Binary

  • Does 8 fit in 13? 13 − 8 = 5 (put a 1 in the 8s place)
  • Does 4 fit in 5? 5 − 4 = 1 (put a 1 in the 4s place)
  • Does 2 fit in 1? No (put a 0 in the 2s place)
  • Does 1 fit in 1? Yes: 1 − 1 = 0 (put a 1 in the 1s place)
  • Read the digits from left to right: 11012
Place Value: 8 4 2 1
Result: 1 1 0 1

So 13 in Decimal = 11012

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