Binary Numbers and Data Representations

Intro

Computers store all information using binary, a base-2 numbering system using only 0 and 1.

• Each digit in binary is called a bit (binary digit).

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Unsigned and Signed Integers

Unsigned Integers

• Only represent non-negative numbers.
• Maximum value depends on the number of bits:

| Bytes | Type     | Bits | Min Value | Max Value                  |
| ----- | -------- | ---- | --------- | -------------------------- |
| 1     | `uint8`  | 8    | 0         | 255                        |
| 2     | `uint16` | 16   | 0         | 65,535                     |
| 4     | `uint32` | 32   | 0         | 4,294,967,295              |
| 8     | `uint64` | 64   | 0         | 18,446,744,073,709,551,615 |

Signed Integers

• Can represent negative numbers, usually using two’s complement.
• Max and min values:

| Bytes | Type  | Bits | Min Value                  | Max Value                 |
| ----- | ----- | ---- | -------------------------- | ------------------------- |
| 1     | int8  | 8    | -128                       | 127                       |
| 2     | int16 | 16   | -32,768                    | 32,767                    |
| 4     | int32 | 32   | -2,147,483,648             | 2,147,483,647             |
| 8     | int64 | 64   | -9,223,372,036,854,775,808 | 9,223,372,036,854,775,807 |
 

Converting Between Binary and Decimal

Binary to Decimal Conversion Table

Binary Number	Decimal Number	Hex Number
0	0	0
1	1	1
10	2	2
11	3	3
100	4	4
101	5	5
110	6	6
111	7	7
1000	8	8
1001	9	9
1010	10	A
1011	11	B
1100	12	C
1101	13	D
1110	14	E
1111	15	F
10000	16	10
10001	17	11
10010	18	12
10011	19	13
10100	20	14
10101	21	15
10110	22	16
10111	23	17
11000	24	18
11001	25	19
11010	26	1A
11011	27	1B
11100	28	1C
11101	29	1D
11110	30	1E
11111	31	1F
100000	32	20
1000000	64	40
10000000	128	80
11111111*	255	FF
100000000	256	100

Notes:

• Binary is the fundamental representation for all data in computers
• Hexadecimal (base-16) is often used as a shorthand for binary because 4 bits = 1 hex digit.

*See Worked Example: Hex ↔ Binary ↔ Decimal for a full 8-bit conversion.

Decimal → Binary

• Divide the decimal number by 2 repeatedly, writing down the remainder each time.
• The binary number is the remainders read from bottom to top.

Example: Convert 13 to binary:

 
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Binary: 1101
 

Binary → Decimal

• Multiply each bit by 2^(position from right, starting at 0) and sum them.

Example: Convert 1101 to decimal:

 
1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 8 + 4 + 0 + 1 = 13
 

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