Converting Between Number Systems

Why Learn Number Conversion?

Number system conversion is a fundamental skill in computer science, networking, and digital electronics. While humans think in decimal, computers operate in binary, and many technical fields use hexadecimal or octal as convenient shorthand for binary data.

Here are real-world scenarios where you need to convert between number systems:

• Web development: CSS color codes use hexadecimal (e.g., #FF5733). To understand or manipulate colors programmatically, you need to convert between hex and decimal RGB values.
• Networking: IP addresses and subnet masks involve binary math. Calculating a subnet range requires converting dotted-decimal notation to binary, performing bitwise operations, and converting back.
• System administration: Unix file permissions (e.g., chmod 755) use octal. Each digit represents three binary bits controlling read, write, and execute permissions.
• Programming: Bitwise operations, bit flags, and low-level data manipulation all require understanding binary and hexadecimal representations.
• Hardware engineering: Memory addresses, register values, and bus data are all expressed in hex or binary.

The good news is that once you understand the underlying principles, conversion becomes mechanical. This guide walks you through every major conversion with step-by-step examples.

Decimal to Binary

To convert a decimal integer to binary, use the repeated division by 2 method. Divide the number by 2, record the remainder (0 or 1), then divide the quotient by 2 again. Continue until the quotient is 0. The binary result is the remainders read from bottom to top.

Example: Convert 156 to binary

Step	Divide	Quotient	Remainder
1	156 ÷ 2	78	0
2	78 ÷ 2	39	0
3	39 ÷ 2	19	1
4	19 ÷ 2	9	1
5	9 ÷ 2	4	1
6	4 ÷ 2	2	0
7	2 ÷ 2	1	0
8	1 ÷ 2	0	1

Reading remainders bottom to top: 156₁₀ = 10011100₂

Verification: 128 + 16 + 8 + 4 = 156 ✓

For decimal fractions: Multiply the fractional part by 2 repeatedly, recording the integer part each time.

Example: Convert 0.8125 to binary

Step	Multiply	Integer Part	Remaining Fraction
1	0.8125 × 2	1	0.625
2	0.625 × 2	1	0.25
3	0.25 × 2	0	0.5
4	0.5 × 2	1	0.0

Reading integer parts top to bottom: 0.8125₁₀ = 0.1101₂

Binary to Decimal

To convert a binary number to decimal, multiply each digit by its positional power of 2 and sum the results. This is also called the positional notation method.

Example: Convert 11010110₂ to decimal

Position	7	6	5	4	3	2	1	0
Binary Digit	1	1	0	1	0	1	1	0
Power of 2	128	64	32	16	8	4	2	1
Value	128	64	0	16	0	4	2	0

Sum: 128 + 64 + 16 + 4 + 2 = 214₁₀

A shortcut for common byte values is to memorize the powers of 2:

Power	2⁰	2¹	2²	2³	2⁴	2⁵	2⁶	2⁷	2⁸	2⁹	2¹⁰
Value	1	2	4	8	16	32	64	128	256	512	1024

For binary fractions, use negative powers of 2. The first digit after the point is 2^-1 (0.5), the second is 2^-2 (0.25), and so on.

Example: Convert 0.1101₂ to decimal

(1 × 0.5) + (1 × 0.25) + (0 × 0.125) + (1 × 0.0625) = 0.5 + 0.25 + 0.0625 = 0.8125₁₀

Decimal to Hexadecimal

Converting decimal to hexadecimal uses the same repeated division method, but dividing by 16 instead of 2. Since hex uses digits 0–9 and letters A–F, remainders of 10–15 must be converted to their hex equivalents.

Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

Example: Convert 4392 to hexadecimal

Step	Divide	Quotient	Remainder	Hex Digit
1	4392 ÷ 16	274	8	8
2	274 ÷ 16	17	2	2
3	17 ÷ 16	1	1	1
4	1 ÷ 16	0	1	1

Reading remainders bottom to top: 4392₁₀ = 1128₁₆

Verification: (1 × 4096) + (1 × 256) + (2 × 16) + (8 × 1) = 4096 + 256 + 32 + 8 = 4392 ✓

Another example with letters: Convert 255 to hexadecimal

Step	Divide	Quotient	Remainder	Hex Digit
1	255 ÷ 16	15	15	F
2	15 ÷ 16	0	15	F

Result: 255₁₀ = FF₁₆ — This is the maximum value of a single byte, which is why white in CSS is #FFFFFF.

Hexadecimal to Decimal

To convert hexadecimal to decimal, multiply each hex digit by its positional power of 16 and sum the results.

Example: Convert 2F1A₁₆ to decimal

Position	Hex Digit	Decimal Value	Power of 16	Result
3	2	2	16³ = 4096	8192
2	F	15	16² = 256	3840
1	1	1	16¹ = 16	16
0	A	10	16⁰ = 1	10

Sum: 8192 + 3840 + 16 + 10 = 12,058₁₀

Common hex values you should know:

Hex	Decimal	Binary	Common Use
0x0A	10	00001010	Line feed (LF)
0x0D	13	00001101	Carriage return (CR)
0x20	32	00100000	Space character
0x30	48	00110000	ASCII '0'
0x41	65	01000001	ASCII 'A'
0x61	97	01100001	ASCII 'a'
0x7F	127	01111111	DEL character
0xFF	255	11111111	Max byte value

Binary to Hexadecimal

Converting between binary and hexadecimal is especially easy because each hex digit maps to exactly 4 binary digits (one "nibble"). To convert, simply group the binary digits into sets of four from the right and replace each group with its hex equivalent.

Example: Convert 101101101001₂ to hexadecimal

Binary Group	0001	0110	1101	0001
Hex Digit	1	6	D	1

Wait — let me regroup correctly. Starting from the right: 1011 0110 1001

Binary Group	1011	0110	1001
Hex Digit	B	6	9

Result: 101101101001₂ = B69₁₆

The reverse works the same way — replace each hex digit with its 4-bit binary equivalent:

Example: Convert 3A7₁₆ to binary

Hex Digit	3	A	7
Binary	0011	1010	0111

Result: 3A7₁₆ = 001110100111₂ (leading zero can be dropped: 1110100111₂)

The Octal Number System

Octal (base-8) uses digits 0 through 7. Each octal digit represents exactly 3 binary digits, making octal-to-binary conversion straightforward — though less commonly used today than hexadecimal.

Binary to Octal: Group binary digits into sets of 3 from the right.

Example: Convert 10110100₂ to octal

Binary Group	010	110	100
Octal Digit	2	6	4

Result: 10110100₂ = 264₈

Decimal to Octal: Use repeated division by 8.

Example: Convert 156 to octal

Step	Divide	Quotient	Remainder
1	156 ÷ 8	19	4
2	19 ÷ 8	2	3
3	2 ÷ 8	0	2

Result: 156₁₀ = 234₈

Octal is still used today primarily in Unix/Linux file permissions. The command chmod 755 file means:

• 7 (owner) = 111₂ = read + write + execute
• 5 (group) = 101₂ = read + execute
• 5 (others) = 101₂ = read + execute

Conversion Cheat Sheet

Here is a complete reference table showing values from 0 to 31 across all four common number systems:

Decimal	Binary	Octal	Hex
0	00000	0	0
1	00001	1	1
2	00010	2	2
3	00011	3	3
4	00100	4	4
5	00101	5	5
6	00110	6	6
7	00111	7	7
8	01000	10	8
9	01001	11	9
10	01010	12	A
11	01011	13	B
12	01100	14	C
13	01101	15	D
14	01110	16	E
15	01111	17	F
16	10000	20	10
17	10001	21	11
18	10010	22	12
19	10011	23	13
20	10100	24	14
21	10101	25	15
22	10110	26	16
23	10111	27	17
24	11000	30	18
25	11001	31	19
26	11010	32	1A
27	11011	33	1B
28	11100	34	1C
29	11101	35	1D
30	11110	36	1E
31	11111	37	1F

Quick conversion rules to remember:

• Binary ↔ Hex: Group/ungroup bits in sets of 4. Each hex digit = 4 binary digits.
• Binary ↔ Octal: Group/ungroup bits in sets of 3. Each octal digit = 3 binary digits.
• Decimal → Any base: Repeated division by the target base, read remainders bottom to top.
• Any base → Decimal: Multiply each digit by its positional weight and sum.
• Hex → Binary → Decimal: Convert hex to binary first, then binary to decimal — often easier than dividing by 16 mentally.

Practicing these conversions until they become second nature will make networking, programming, and system administration tasks significantly easier.