In mathematics, a binary number is, simply put, a digit of the binary numeral system. This is a system widely used today, despite us being practically unaware of it. Virtually all electronic devices, ranging from the cheapest calculator to the most expensive computer, rely on a form of the binary system as an integral part of their internal workings.

There has been historic proof that ancient civilizations employed the binary system to some extent, including 9th century China. This specific incarnation of it was researched by Gottfried Wilhelm Leibniz, a 17th century German philosopher and mathematician. A contemporary of Sir Isaac Newton, he is credited with a number of mathematical innovations, including development of integral calculus and the refinement of the binary system. For this, he is sometimes referred to as the first computer scientist. In his work, Leibniz saw and interpreted the binary system as proof for existence of God, or rather, the creation of something out of nothing.

In the 19th century, George Bool, a British mathematician, published The Laws of Thought, laying foundations for what we now call boolean algebra in which the binary system is employed in logical context, instead of a mathematical one.

The first computer to employ the binary system was constructed by George Robert Stibitz, an American researcher, in 1937. The Model K („K“ stands for „kitchen“, where he completed it), was replaced in 1939 by the Complex Number Computer, which was able to calculate complex numbers. It is also the first computer be able to be used remotely, using the teletype (another contemporary device), and a standard phone line. Binary system today is the foundation of all computer systems. On its most basic level, the computer sees everything as zeros and ones, or rather, voltage and lack of thereof.

The math behind conversion from our real world decimal system to binary is quite simple, and we use an algorithm that lets us convert a rumen easily. For example, let's convert the number 81 to binary. Since the base (the number of different digits used in a numeral system) of the binary system is 2, we will be dividing our starting number by 2, until we reach 0, after which there's no more point in dividing it. After each division, we'll write down the remainder.

81/2=40 1

In this step, we divide 81 by 2. When converting from decimal to binary, the only two possible remainders are either 0 or 1.

40/2=20 0

Since we're dividing an even number by another even number, there's no remainder.

20/2=10 0

10/2=5 0

5/2=2 1

2/2=1 0

1/2=0 1

Once we reach this last step, we look at our remainders, and write it down in ascending order: 1010001. This number is what we were looking for, 81 in it's binary form. The common way of expressing this mathematically is 8110=10100012, where the indexes represent the numeral base, also known as radix.

To convert from binary to decimal, we'll use a different algorithm. Let's convert the number 1110110012 to it's decimal equivalent. We'll start off by writing the number in the opposite order, from right to left, and mark their positions starting from 0.

0 1 2 3 4 5 6 7 8

1 0 0 1 1 0 1 1 1

Now, the idea is to multiply the digits by the base to the power of the position and then add them all together. It may sound a little complicated, but is in fact rather simple. The first digit is 1, which we multiply by 20 (base to the power of the position)
The entire string would go like this:

1x20 + 0x21 + 0x22 + 1x23 + 1x24 + 0x25 + 1x26 + 1x27 + 1x28 =

1 + 0 + 0 + 8 + 16 + 0 +64 + 128 + 256 = 473

Which means that:

1110110012 = 47310

Binary numbers are also easily convertible to any numeral system whose base is 2n. The reason the octal and hexadecimal systems are also widely used in computing is because their bases conform to this rule, being 8 (23) and 16 (24) respectively. So, if we would want to convert the number 110000110100 to octal, we would simply make group the digits 3 by 3 and translate them to the octal system with the pattern bellow.

Octal Binary

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

Which means our number (110)(000)(110)(100)2 equals 60648. Using the same method, we can turn octal numbers to binary as well. A very similar system is used to convert binary numbers to their hexadecimal values.

To summarize, the binary system is a numeral system with the base 2, meaning it uses two different digits: 0 and 1. The binary system is the foundation of computing and information technology as we know it, and we've presented it's history, significance, and use.