Binary number

system that represents numeric values using two symbols; 0 or 1

The binary numeral system is a base 2 number system. It is called base 2 because each digit can only be 2 possible values: 0 and 1. Therefore, every digit's place value is two times the previous digit.

Computers work in binary, because it is the simplest way to store information using electricity. A wire can have no electricity for 0, or some electricity for 1.

Decimal
number
Binary
number
Binary
expansion
0 0 zero
1 1 one
2 10 two
3 11 two + one
4 100 four
5 101 four + one
6 110 four + two
7 111 four + two + one
8 1000 eight
9 1001 eight + one
10 1010 eight + two
11 1011 eight + two + one
12 1100 eight + four
13 1101 eight + four + one
14 1110 eight + four + two
15 1111 eight + four + two + one

Binary arithmetic

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Arithmetic is a way of adding together two or more binary numbers. There are four rules in binary arithmetic. They are:

0 + 0 = 0
1 + 0 = 1
1 + 1 = 10 (2)
1 + 1 + 1 = 11 (3)

This is because in binary there are only two digits; 0 and 1. Because of this, the number two and three have to be represented in some other way. This is how the binary value for three is calculated:

Column Decimal Value Binary
1 2 1
2 1 1

This shows that the binary value would be 11.

History

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The binary numeral system, in a manuscript by Gottfried Wilhelm Leibniz, 1697.
 
A page from „Explication de l’Arithmétique Binaire“, by Leibniz, 1703.

John Leslie

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In 1817, John Leslie (a Scottish mathematician) suggested that primitive societies may have evolved counting with objects (like pebbles) before they had even words to describe the total number of objects involved. The next step in the evolution of counting would have been the discovery that this pile of objects could be reduced into two piles of equal measures (leaving either 0 objects left over or just a remainder of 1).

This remainder (odd = 1 or even = 0) would then be recorded and one of the piles removed whilst the second pile was then further divided into two sub piles.

If you record the remainder left over after the original pile has been divided in two and continue repeating this process; of sub dividing one of the remaining piles into half and then removing one of those piles and continue by subdividing the remaining pile into two piles you will ultimately be left with just either 2 or 3 objects.

If you record the remainder left over (odd = 1 or even = 0) at the end of each reduction you will eventually be left with a tally record of 1's and 0's which will be the binary representation of your original pile of objects. So instead of representing your original pile of objects with a repeating number or marks or tokens (which for large numbers could be quite long) you have reduced your pile of objects into more compact binary number.

If you need to recover the original number of objects from this summarised binary number it is easy enough to do; by simply starting with the first tally mark and then doubling it and adding one if the next binary number contains a 1 and then continuing the process until the end of the binary number is reached. So, binary counting may be both the oldest and the most modern method of counting.

Applications

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Binary was invented by many people, but the modern binary number system is credited to Gottfried Leibniz in 1679, a German mathematician. Binary has been used in nearly everything electronic; from calculators to supercomputers. Machine code is binary digits.

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Translation

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You can translate binary to normal numbers using