In the class, we learned about the strange base-60-system of Babylon, and I was wondering were there any other counting system that seems unfamiliar to us.

Because I am taking a programming class, the first thing that came into my mind was the binary system. Binary numbers represent values using only two different symbols: 0(zero) and 1(one). This system seems easier than base-10 system, because we only need to remember two symbols to express all integers. For instance, the first 10 integers in the decimal system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) can be expressed as : “ 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001.”

- Origin of binary numbers

Although occupying more space, the expression of numbers in the binary system seems easier than in the decimal system. Then I am wondering who first invented it? It is said Gottfried Leibniz*,* a German mathematician and philosopher who is famous for the inventing of Calculus, first discover the modern binary number system and it appears in his article *“Explanation of the Binary Arithmetic”* . Leibniz also indicated that the ancient ruler of China* Fuxi* first invented the binary system in his work — *“I Ching”*; in *“I Ching”, *the binary numbers are being used to divine the fate of ancient Chinese people, for those people believe that the mysterious secrets of the universe are all in these simple numbers of signs.

- The arithmetic of binary numbers

Like the decimal system, binary numbers also have their arithmetic.

- Addition

Addition is the simplest operation in the binary system. Adding two single-digit binary numbers is relatively easy, like this:

0 + 0 → 0

0 + 1 → 1

1 + 0 → 1

1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )

Here when adding two 1s, we carry the value divided by 2, and add the quotient to it the left-next positional value, so multiple-digits number addition is:

0 1 1 0 1

+ 1 0 1 1 1

—————–

= 1 0 0 1 0 0

- Subtraction

The subtraction of binary numbers is the inverse operation of addition, when two single-digit number doing a subtraction, like this :

0 − 0 → 0

0 − 1 → 1, borrow 1(from the left-next position)

1 − 0 → 1

1 − 1 → 0

So, likewise, the multiple-digits numbers’ subtraction are like this:

* * * * (starred columns are borrowed from)

1 1 0 1 1 1 0

− 1 0 1 1 1

——————

1 0 1 0 1 1 1

### 3)Multiplication

Multiplication in binary numbers is simpler than in the decimal system, for two number A and B, there are only two rules:

- If the digit in Bis 0, the partial product is also 0
- If the digit in B is 1, the partial product is equal to A

For example, the binary numbers 1011 and 1010 are multiplied as follows:

1 0 1 1 (A)

× 1 0 1 0 (B)

———–

0 0 0 0 ← Corresponds to the rightmost ‘zero’ in B

1 0 1 1 ← Corresponds to the next ‘one’ in B

0 0 0 0

+ 1 0 1 1

—————–

= 1 1 0 1 1 1 0

And for division, it is the inverse operation of multiplication.

- Transfer between binary number and decimal number

How to transfer a binary number into a decimal number? For example:

11011(2) = 1 * 2^4 + 1*2^3+0*2^2+1*2^1+1=27(10)

And the inverse transfer is to count the power of 2s in a decimal number, like:

36(10) = 1*2^5+ 0*2^4 +0*2^3+1*2^2+0*2^1+0*2^0= 100100(2)

- The application of binary numbers

The reason why I introduce the binary numbers is that they are the base of modern science, especially the computer science. The basic element of a computer is the logical circuit, which only has two basic situations: 0 for switch off, and 1 for switch on. As the old saying: less is more. The binary system is coincidentally perfectly fitting the feature of the logical circuit(0 for no, and 1 for yes). And the logical circuit led to the invent of computer. For example, when calculating, the computer will translate the numbers into binary form and do the operations, and then transfer the answer back to decimal number like it shown above.

- Conclusion

It is unbelievable when you think of the powerful computer is based on the binary system, and considering the huge works computer have done so far, we can say that the binary system is the key of modern science and technologies. Even when I am typing this article, the binary numbers keep working in my computer.

Reference:

http://en.wikipedia.org/wiki/Binary_number

http://en.wikipedia.org/wiki/Logic_gate#Symbols

http://www.ask.com/technology/computers-use-binary-number-system-33ebc182c16b88f