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People fall into 10 groups, those that get binary and everyone else!

The full joke goes: "There are 10 types of people in this world. Those who understand binary and those who don't." If you are puzzed by the statement, read on and become au fait with binary.

We Normally Count in Tens

The most common form of counting and numbers involves the ten digits from 0 to 9. Counting goes thus: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 etc. ad infinitum. When the count gets to 9 there are no more digits to use for the next number, ten, so two digits are used, with the first one representing the number of tens to use and second one the remaining count. Hence 10 is 1 lot of tens + 0, 11 is 1 lot of tens + 1, 12 is 1 lot of tens + 2, etc. Likewise 20 is 2 lots of tens + 0, 21 is 2 lots of ten plus 1 etc. We call this counting in base ten because all numbers are made up of lots of tens, i.e. ten is the base for all the numbers. Base ten is also known as denary or decimal. When we get to 99 and count one more we need to add another number and have a three digit number, so 100 is 1 lot of ten lots of ten + 0 lots of ten + 0 (it is ten lots of ten because we know we have counted to ten, ten times when we have counted to 99 and counted one more). Ten lots of ten has the name a hundred, much easier to say and remember. So 100 is 1 hundred + 0 lots of tens + 0. And their are names to help with bigger numbers, thousand (ten lots of one hundred) and million (one thousand lots of one thousand). Base ten, or decimal, is our normal counting system and we have ten digits on our hands (four fingers and one thumb on each hand), maybe that's were we got it from.

We Can Count in Different Numbers

If only the digits 0 to 7 were available counting would work the same way: 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 etc. There is no 8 or 9 so we have to move to the next digit earlier when we get to what would have been eight in our normal counting. Thus 10 is no longer representing our normal ten, it is 1 lot of eights + 0, and 11 is 1 lot of eights + 1, 12 is 1 lot of eights + 2, etc. This is counting in eights, base eight, also known as octonary, or octal. So a number like 23 is 2 lots of eight plus 3 (2×8+3=19 in decimal). Maybe if humans had only eight digits on their hands octal would have been our normal counting system.

Counting with the Digits 0 and 1

After starting to count using the digits 0 to 9, for base ten, it was shown that counting works with fewer digits, 0 to 7, base 8. The same principles apply if more digits are removed. What about just two digits, just 0 and 1, base 2. The count is 0 then 1, there is no 2 so we add another digit that represents the number of twos, so 10 is 1 lot of twos + 0 = 2 decimal. 11 is one lot of twos + 1 = 3 decimal. Counting with just the digits 1 and 0 is really quite easy, when adding a 1 to a 1 you carry a one, lengthening the number by another 1 digit if the number is all 1's:

Counting in Base 2 with 1 and 0
0 + 1 = 1
1 + 1 Adding 1 to 1, set 0 carry the 1, adding another digit = 10
10 + 1 = 11
11 + 1 Adding 1 to 1, all 1's set to 0, carry 1 to end adding another digit = 100
100 + 1 = 101
101 + 1 Adding 1 to 1, carry the 1 = 110
110 + 1 = 111
111 + 1 Adding 1 to 1, all 1's set to 0, carry 1 to end adding another digit = 1000
1000 + 1 = 1001
1001 + 1 Adding 1 to 1, carry 1 = 1010
1010 + 1 = 1011
etc.

So the count goes 0 1 10 11 100 101 110 111 1000 1001 etc. The method of counting in base 2, using just the digits 0 and 1, is called binary. Looking at the start of the counting sequence in binary it was seen that 0 binary = 0 base ten, 1 binary = 1 base 10, 10 binary = 2 base 10. Now the quip "There are 10 types of people in this world. Those who understand binary and those who don't." would make sense, since 10 is binary for 2.

Bits

Each digit in a binary number is called a bit. Thus the binary number 10011 (19 in decimal) is made up of five bits. A single bit takes the value 0 or 1. Sometimes physical items are used to represent or illustrated a binary value, a concept important when writing computer programs. A light can be off or on. With off representing 0 and on representing 1. Other items can represent such two states such as a gate or door which is closed (0) or open (1), the answer to a question is wrong (0) or right (1), or a statement (the glass is full) is false (0) or true (1).

Lights representing 19 binary

Each bit in a binary number represents 2x (twice) the value, in decimal, of the number before it with the first number representing 1:

Values of Binary Digits
Bit Position 9 8 7 6 5 4 3 2 1
Decimal value when set 256 128 64 32 16 8 4 2 1
Equals 2x its left neighbour 2×128 2×64 2×32 2×16 2×8 2×4 2×2 2×1 NA = 1
Equivalent To 28 27 26 25 24 23 22 21 20

Notice the value of a bit is 2 raised to the power of the bits' position minus 1. Thus converting from binary to decimal is done by adding powers of 2.

Converting Binary to Decimal
Bits 1 0 1 0 0 1 1 1
Position 8 7 6 5 4 3 2 1
Value is x2pos-1 1x27 0x26 1x25 0x24 0x23 1x22 1x21 1x20
Equals 1x128 0x64 1x32 0x16 0x8 1x4 1x2 1x1
Total 128 + 0 + 32 + 0 + 0 + 4 + 2 + 1 = 167

Converting from decimal to binary is straightforward as well. Since it involves powers of two the number to be converted is divided by two and the remainder written down. This is done until no more division is possible. The written down remainders are the binary bits in reverse:

Converting Decimal to Binary
167 / 2 = 83 remainder 1
83 / 2 = 41 remainder 1
41 / 2 = 20 remainder 1
20 / 2 = 10 remainder 0
10 / 2 = 5 remainder 0
5 / 2 = 2 remainder 1
2 / 2 = 1 remainder 0
1 / 2 = 0 remainder 1

Reading the resulting bits backwards thus 10100111 = 167.

After reading this article I hope you now know which of the 10 groups into which you fall.

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