Binary Counting ("Base 2")

Computers do not count with the same counting system we use. Humans have ten fingers, so we use a base-10 counting system--that is, 10 digits. A "digit" is a single-character number; the ten digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Computers use switches to keep information. A switch can be off or on. On is 1; off is 0. SO computers only use those two digits. One number--a 1 or a 0--is called a binary digit, or a "bit" for short.

When we count, we start with a single digit, going from 0 to 9. After 9, there are no more digits. So we have to use two digits. Each digit represents a "column," as shown in the chart below. In the number "10," the "1" is in the "tens" column, and equals "1 x 10"; the "0" is in the "ones" column, and equals "0 x 1."

As an example, take the number 256:

Start at the right, and move to the left

 Millions Hundred Thousands  Ten Thousands   Thousands  Hundreds  Tens  Ones
1,000,000 100,000 10,000 1,000 100 10 1
106 105 104 103 102 101
 0  0  0  0  0  0  0
 1  1  1  1  1  1  1
 2  2  2  2  2  2  2
 3  3  3  3  3  3  3
 4  4  4  4  4  4  4
 5  5  5  5  5  5  5
 6  6  6  6  6  6  6
 7  7  7  7  7  7  7
 8  8  8  8  8  8  8
 9  9  9  9  9  9  9

Therefore, the number "256" means that you have 2 hundreds, 5 tens, and 6 ones. Each digit has ten possibilities, so with each new digit, you multiply by ten. A 2-digit number has 100 possibile combinations. A 3-digit number has 1000 combinations, and so on.


However, if you only have two numbers instead of ten, your counting has to look like this:

 Start at the right, and move to the left

 five hundred and twelves  two hundred and fifty- sixes  one hundred and twenty- eights  sixty- fours thirty- twos  sixteens  eights  fours  twos  ones
512 256 128 64 32  16  8  4  2  1
29 28 27 26 25 24 23 22
21
 0  0  0  0  0  0  0  0  0  0
 1  1  1  1  1  1  1  1  1  1

Notice the numbers: 8, 16, 32, 64, 128, 256, 512. These are numbers seen very often with computer memory.

To write the number 7, you would write "111"--that is, 1 four, 1 two, and 1 one. To write the number 8, you would write "1000"--1 eight, and 0 fours, twos and ones. The number 256, therefore, is "10000000."

Notice that the second column is the "base," and all other columns are powers of the base--base2, base3, base4, etc.

Here is an example of counting from one to ten in binary:

 binary    base 10
 0    0
 1   1
 10   2
 11    3
 100    4
 101    5
 110    6
 111    7
 1000    8
 1001    9
 1010    10

Each digit has two possibilities, so with each new digit, you multiply by two. For example, a 2-digit binary number has four possible combinations (00,01,10, and 11). A 3-digit number has eight combinations; a 4-digit number has 16 combinations, and so on.


 

Try using this binary - base 10 conversion app to see how binary numbering works: