The Binary System

To understand how computers work, we must understand how a number system with only ones and zeroes works.

[...] any information that can be reduced to a choice among two or more possibilities can be expressed using bits. (Charles Petzold)

Key Points

The number system we use in our daily lives is the system to the base 10, or the decimal system (decimus is Latin for "tenth").
The fact that we use the decimal system does not mean other systems do not exists. In fact, every other number system to other bases exists as well, and we can convert numbers between different systems.
The word "decimal" or base ten implies that in that system, we have 10 symbols to represent numbers (0 - 9). Accordingly, the number system to the base 8, which is the octal system, has only eight symbols (0 - 7).
Computers use the binary system (bīnārius is Latin for "consisting of two"), which has only two symbols: 0 and 1.
Regardless of how many symbols we have in a given system, representing numbers in a system follows the same approach. In a system to the base $n$ , the first digit (also called least significant digit) represents the number $n^0$ , which is always one. The second stands for $n^1$ , the one after that for $n^2$ and so forth. So the octal number 10 is equal to the decimal number 8: $1 \times 8^1 + 0 \times 8^0$ , and the octal number 110 would be the decimal number 72: $1 \times 8^2 + 1 \times 8^1 + 0 \times 8^0$
We can apply the same approach for the binary system. The binary number 1100 is equivalent to the decimal number 12: $1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 1 \times 8 + 1 \times 4 = 12$

Material

Title

Link

PDF

Slides "The Binary System"

Video "Number Systems Introduction - Decimal, Binary, Octal, Hexadecimal

Text "Our Ten Digits" (8 pages)

Text "Alternatives to Ten" (18 pages)

For students from the , all texts are available through our team room.

References

[1] Petzold, Charles. Code: The Hidden Language of Computer Hardware and Software. Paperback ed, Microsoft Press, 2000.

[2] Scott, J. Clark. But How Do It Know?: The Basic Principles of Computers for Everyone. John C. Scott, 2009.

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Key Points

The number system we use in our daily lives is the system to the base 10, or the decimal system (decimus is Latin for "tenth").

The fact that we use the decimal system does not mean other systems do not exists. In fact, every other number system to other bases exists as well, and we can convert numbers between different systems.

The word "decimal" or base ten implies that in that system, we have 10 symbols to represent numbers (0 - 9). Accordingly, the number system to the base 8, which is the octal system, has only eight symbols (0 - 7).

Computers use the binary system (bīnārius is Latin for "consisting of two"), which has only two symbols: 0 and 1.

Regardless of how many symbols we have in a given system, representing numbers in a system follows the same approach. In a system to the base $n$ , the first digit (also called least significant digit) represents the number $n^0$ , which is always one. The second stands for $n^1$ , the one after that for $n^2$ and so forth. So the octal number 10 is equal to the decimal number 8: $1 \times 8^1 + 0 \times 8^0$ , and the octal number 110 would be the decimal number 72: $1 \times 8^2 + 1 \times 8^1 + 0 \times 8^0$

We can apply the same approach for the binary system. The binary number 1100 is equivalent to the decimal number 12: $1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 1 \times 8 + 1 \times 4 = 12$