Number System

Number Representation

A number can be represented by

$$X_{n} X_{n - 1} \dots X_1 X_0 X_{-1} \dots X_{-m} = \sum^{n}_{i=-m}{X_i} \times 10^{i}$$

Where $X$ is base or radix.

Let $b$ be the base, then $X \in [0, b)$

Common Bases

Name	Base	Postfix
Binary Number	$2$	$B$
Octal Number	$8$	$Q$ or $O$
Decimal Number	$10$	$D$
Hexadecimal Number	$16$	$H$

Number Conversions

Convert to Decimal Numbers

Suppose there is a number in base $b$, which looks like $a_n a_{n-1} \dots a_1 a_0 . a_{-1} a_{-2} \dots a_{-m}$, then the formula of converting it to a decimal number is:

$$\sum_{i=-m}^{n}{a_i b^i}$$

Convert A Decimal Number to Other Bases

Suppose there is a decimal number n, and it needs to be converted to a number based b. Then the process is:

1. Calculate $a = n \mod b$.
2. Append $a$ to the end of result.
3. Let $n = n / b$
4. Repeat steps 1 to 3 until $n = 0$.

Conversion Between Non-decimal Numbers

Convert a number to binary number, and then convert the binary number to the target number.

Non-numeric Data Representation

Binary Coded Decimal (BCD) Code

The range of a BCD digit is 0000B to 1001B, also $0 - 9$ decimal.

It has two types, packed BCD and unpacked BCD.

• Packed BCD stores two digits per byte, which is better for storage.
• Unpacked BCD stores one digit per byte, which is easy for computer to access data.

ASCII Code

The range of ASCII code is from $0$ to $127$, while extended ASCII code is range from $0$ to $255$.

Units

• Bit ($0$ or $1$)
• Byte (8 bits)
• Word (16 bits)
• Double word (32 bits)