Binary Digits

A Binary Digit can only be 0 or 1

Binary Number

A Binary Number is made up Binary Digits.

 

In the computer world "binary digit" is often shortened to the word "bit"

More Than One Digit

So, there are only two ways we can have a binary digit ("0" and "1", or "On" and "Off") ... but what about 2 or more binary digits?

Let's write them all down, starting with 1 digit (you can test it yourself using the switches):

2 ways to have one digit ...
0
1
... 4 ways to have two digits ...
0 0 00
1 01
1 0 10
1 11
... 8 ways to have three digits ...
0 0 0 000
1 001
1 0 010
1 011
1 0 0 100
1 101
1 0 110
1 111
... and 16 ways to have four digits.
0 0 0 0 0000
1 0001
1 0 0010
1 0011
1 0 0 0100
1 0101
1 0 0110
1 0111
1 0 0 0 1000
1 1001
1 0 1010
1 1011
1 0 0 1100
1 1101
1 0 1110
1 1111

And, in fact, we have created the first 16 binary numbers:

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

This is useful! To remember the sequence of binary numbers just think:

  • "0" and "1" {0,1}
  • then repeat "0" and "1" again but with a "1" in front: {0,1,10,11}
  • then repeat those four with "1"s as a third digit: {0,1,10,11,100,101,110,111}
  • and so on!

(It is also how we count using decimal numbers, but we then also use 2, 3 , 4, 5, 6, 7, 8 or 9.)

Now find out how to use Binary to count past 1,000 on your fingers:

Also have a Play with 4 different drums.

Binary Digits ... They Double!

Also notice that each time we add another binary digit we double the possibilities.

Why double? Because we to take all the previous possible positions and match them with a "0" and a "1" like above.

  • So just one binary digit has 2 possible positions
  • Two binary digits have 4 possible positions
  • Three have 8 possible positions
  • Four have 16 possible positions
  • Five have 32 possible positions
  • Six have 64 possible positions
  • etc.

Using exponents, this can be shown as:

No of Digits Formula Settings
1 21 2
2 22 4
3 23 8
4 24 16
5 25 32
6 26 64
etc... etc... etc...

Example: when you have 50 binary digits (or even 50 things that can only have two positions each), how many different ways is that?

Answer 250 = 2 × 2 × 2 × 2 ... (fifty of these) = 1,125,899,906,842,624

So, a binary number with 50 digits could have 1,125,899,906,842,624 different values.

Or to put it another way, it could show a number up to 1,125,899,906,842,623 (note: this is one less than the total number of values, because one of the values is 0).

Chess Board

chess board

There is an old Indian legend about a King who was challenged to a game of chess by a visiting Sage. The King asked "what will be the prize if you win?".

The Sage said he would simply like some grains of rice: one on the first square, 2 on the second, 4 on the third and so on, doubling on each square. The King was surprised by this humble request.

Well, the Sage won, so how many grains of rice should he receive?

On the first square: 1 grain, on the second square: 2 grains (for a total of 3) and so on like this:

Square Grains Total
1 1 1
2 2 3
3 4 7
4 8 15
10 512 1,027
20 524,288
1,048,575
30 53,6870,912
1,073,741,823
64 ???
???

By the 30th square you can see it is already a lot of rice! A billion grains of rice is about 25 tonnes (1,000 grains is about 25g ... I weighed some!)

Notice that the Total of any square is 1 less than the Grains on the next square (Example: square 3's total is 7, and square 4 has 8 grains). So the total of all squares is a formula: 2n−1, where n is the number of the square. For example, for square 3, the total is 23−1 = 81 = 7

So, to fill all 64 squares in a chess board would need:

264−1 = 18,446,744,073,709,551,615 grains (460 billion tonnes of rice),

many times more rice than in the whole kingdom.

So, the power of binary doubling is nothing to be taken lightly (460 billion tonnes is not light!)

(By the way, in the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt immediately but can pay him over time, just serve rice to pilgrims every day until the debt is paid off.)

Hexadecimal

Lastly, I would like to tell you about the special relationship between Binary and Hexadecimal.

There are 16 Hexadecimal digits, and we already know that 4 binary digits have 16 possible values. Well, this is exactly how they relate to each other:

Binary: 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
Hexadecimal: 0 1 2 3 4 5 6 7 8 9 A B C D E F

So, when people use computers (which prefer binary numbers), it is a lot easier to use the single hexadecimal digit rather than 4 binary digits.

For example, the binary number "100110110100" is "9B4" in hexadecimal. I know which I would prefer to write!