Number System Converter: Binary, Decimal, Hexadecimal, Octal — All-in-One Guide

Number Systems Explained: Binary, Decimal, Hex, Octal

Computers operate using binary — a base-2 number system with only two digits: 0 and 1. But humans prefer decimal (base 10, digits 0-9), programmers use hexadecimal (base 16, digits 0-9 and A-F), and Unix systems still use octal (base 8, digits 0-7) for file permissions.

Why multiple number systems exist:

System	Base	Digits	Where It's Used
Binary	2	0, 1	Computer hardware, machine code, digital circuits
Decimal	10	0-9	Everyday human counting, mathematics
Hexadecimal	16	0-9, A-F	Memory addresses, web colours, debugging, assembly
Octal	8	0-7	Unix/Linux file permissions (chmod), legacy systems

The relationship between systems:

• 1 hex digit = 4 binary bits
• 1 octal digit = 3 binary bits
• 1 byte (8 bits) = 2 hex digits = approximately 2.67 octal digits

This relationship means conversion between these systems is straightforward — each system is a different grouping of the same underlying binary data.

Binary ↔ Decimal Converter — The Foundation

The Binary to Decimal Converter is the most fundamental number system converter. Binary is the language of computers — every file, image, and program on your computer is ultimately stored as binary.

How binary to decimal conversion works:

Each binary digit (bit) represents a power of 2. Starting from the rightmost digit (position 0), multiply each digit by 2 raised to its position, then sum all results.

Binary	Calculation	Decimal
1	1 × 2⁰ = 1	1
10	(1 × 2¹) + (0 × 2⁰) = 2 + 0	2
1010	(1×8)+(0×4)+(1×2)+(0×1) = 8+0+2+0	10
11111111	(1×128)+(1×64)+(1×32)+(1×16)+(1×8)+(1×4)+(1×2)+(1×1)	255

How decimal to binary conversion works:

Repeatedly divide the decimal number by 2, recording remainders from bottom to top.

Example: 13 decimal to binary

Division	Quotient	Remainder
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading remainders from bottom to top: 13 decimal = 1101 binary.

Common binary values to memorise:

Decimal	Binary	Use Case
0	0	Zero
1	1	One
2	10	Binary 10
4	100	First power of 2
8	1000	Byte start
16	10000	Half byte
32	100000	ASCII space
64	1000000	@ symbol
128	10000000	High bit
255	11111111	Maximum byte value

Use our Binary Decimal Converter to instantly convert any number between these two systems.

Binary ↔ Hexadecimal Converter — For Programmers

The Binary to Hexadecimal Converter is essential for programmers working with memory addresses, colour codes, and assembly language. Hexadecimal is the standard human-readable representation of binary because it is 4× more compact.

Why hex is so useful:

A single byte (8 bits) is exactly represented by two hex digits. FF = 255, 00 = 0. This perfect alignment makes hex the natural choice for low-level programming.

How binary to hexadecimal conversion works:

1. Group binary digits into sets of 4 from the right
2. Convert each 4-bit group to its hex digit (0-9, A-F)

4-bit Binary	Hex Digit
0000	0
0001	1
0010	2
0011	3
0100	4
0101	5
0110	6
0111	7
1000	8
1001	9
1010	A
1011	B
1100	C
1101	D
1110	E
1111	F

Example: Binary 11011010 to hex

Step	Binary	Hex
Group 4 bits	1101 1010	—
First group	1101	D
Second group	1010	A
Result	—	DA

Where hex is used in real life:

Use Case	Example
Web colours	#FF5733 (red=FF, green=57, blue=33)
Memory addresses	0x7FFD8A3F2B1C
MAC addresses	00:1A:2B:3C:4D:5E
IPv6 addresses	2001:0db8:85a3:0000:0000:8a2e:0370:7334
Debugging	Values in registers shown as hex

Use our Binary Hexadecimal Converter to convert between these systems instantly.

Binary ↔ Octal Converter — For Unix/Linux Permissions

The Binary to Octal Converter is essential for understanding Unix/Linux file permissions. While less common than hex, octal is still used daily by system administrators.

How binary to octal conversion works:

1. Group binary digits into sets of 3 from the right
2. Convert each 3-bit group to its octal digit (0-7)

3-bit Binary	Octal Digit
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

Example: Binary 111010 to octal

Step	Binary	Octal
Group 3 bits	111 010	—
First group	111	7
Second group	010	2
Result	—	72

Where octal is used today — chmod file permissions:

Octal	Binary	Permission	Meaning
0	000	---	No permissions
1	001	--x	Execute only
2	010	-w-	Write only
3	011	-wx	Write and execute
4	100	r--	Read only
5	101	r-x	Read and execute
6	110	rw-	Read and write
7	111	rwx	All permissions

Common chmod examples:

Command	Binary	Meaning
chmod 755	111 101 101	Owner:rwx, Group:r-x, Others:r-x (directories)
chmod 644	110 100 100	Owner:rw-, Group:r--, Others:r-- (files)
chmod 777	111 111 111	Everyone has full access (security risk)

Use our Binary Octal Converter to convert between binary and octal for chmod calculations.

Decimal ↔ Hexadecimal Converter — For Web Colours

The Decimal to Hexadecimal Converter is essential for web developers working with RGB colour codes. Every colour in HTML/CSS can be expressed as a hex code.

How decimal to hexadecimal conversion works:

Repeatedly divide the decimal number by 16, recording remainders (0-15, with 10-15 as A-F).

Example: 255 decimal to hex

Division	Quotient	Remainder	Hex Digit
255 ÷ 16	15	15	F
15 ÷ 16	0	15	F

Read remainders from bottom to top: 255 decimal = FF hex.

RGB colour conversion:

RGB colours are three decimal values (0-255 each). Convert each to hex and combine with #.

Colour	RGB (Decimal)	Hex Code
Red	255, 0, 0	#FF0000
Green	0, 255, 0	#00FF00
Blue	0, 0, 255	#0000FF
White	255, 255, 255	#FFFFFF
Black	0, 0, 0	#000000
Yellow	255, 255, 0	#FFFF00
Purple	128, 0, 128	#800080

Common hex values to memorise:

Decimal	Hex	Use
0	00	Minimum colour value
51	33	Dark shade
102	66	Medium-dark
153	99	Mid shade
204	CC	Medium-light
255	FF	Maximum colour value

Use our Decimal Hexadecimal Converter to convert RGB values to hex codes for web design.

Decimal ↔ Octal Converter — For chmod Commands

The Decimal to Octal Converter is essential for system administrators setting file permissions. The chmod command uses octal numbers (0-7) to represent permission sets.

How decimal to octal conversion works:

Repeatedly divide the decimal number by 8, recording remainders (0-7).

Example: 493 decimal to octal

Division	Quotient	Remainder
493 ÷ 8	61	5
61 ÷ 8	7	5
7 ÷ 8	0	7

Read remainders from bottom to top: 493 decimal = 755 octal.

Common chmod values in decimal and octal:

Decimal	Octal	Permissions
493	755	rwxr-xr-x (directories)
420	644	rw-r--r-- (files)
511	777	rwxrwxrwx (all access)
384	600	rw------- (private)
292	444	r--r--r-- (read-only)

How to read chmod permissions:

• First digit (owner): 7 = rwx (read, write, execute)
• Second digit (group): 5 = r-x (read, execute)
• Third digit (others): 5 = r-x (read, execute)

Use our Decimal Octal Converter to convert decimal numbers to octal for chmod commands.

Hexadecimal ↔ Octal Converter — Cross-System Conversion

The Hexadecimal to Octal Converter helps when working across systems that use different number systems — debugging memory addresses (hex) while setting file permissions (octal).

How to convert between hex and octal:

The most reliable method is to go through decimal first:

1. Hex → Decimal → Octal
2. Octal → Decimal → Hex

Example: Hex DEAD to octal

Step	Calculation	Result
Hex DEAD to decimal	D=13, E=14, A=10, D=13	13×4096 + 14×256 + 10×16 + 13 = 57,005
Decimal 57,005 to octal	Repeated division by 8	157255

Hex DEAD = Octal 157255

Why convert between hex and octal?

Scenario	Need
Debugging memory addresses (hex) while checking file permissions (octal)	Cross-reference
Legacy code that uses octal while modern systems use hex	Compatibility
Learning number system relationships	Education

Use our Hex Octal Converter to convert between these systems instantly.

Complete Conversion Reference Table (All 4 Systems)

This table shows the same number represented in all four number systems simultaneously — a powerful reference for understanding the relationships.

Numbers 0-31

Decimal	Binary	Hexadecimal	Octal
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	8	10
9	1001	9	11
10	1010	A	12
11	1011	B	13
12	1100	C	14
13	1101	D	15
14	1110	E	16
15	1111	F	17
16	10000	10	20
17	10001	11	21
18	10010	12	22
19	10011	13	23
20	10100	14	24
21	10101	15	25
22	10110	16	26
23	10111	17	27
24	11000	18	30
25	11001	19	31
26	11010	1A	32
27	11011	1B	33
28	11100	1C	34
29	11101	1D	35
30	11110	1E	36
31	11111	1F	37

Important Computer Numbers

Decimal	Binary	Hexadecimal	Octal	Significance
255	11111111	FF	377	Maximum byte value
256	100000000	100	400	2⁸
1024	10000000000	400	2000	2¹⁰ (1KB)
4096	1000000000000	1000	10000	2¹² (4KB page)
65535	1111111111111111	FFFF	177777	16-bit maximum
16777215	111111111111111111111111	FFFFFF	77777777	24-bit colour max

Powers of 2 Reference

Power of 2	Decimal	Binary	Hexadecimal	Octal
2⁰	1	1	1	1
2¹	2	10	2	2
2²	4	100	4	4
2³	8	1000	8	10
2⁴	16	10000	10	20
2⁵	32	100000	20	40
2⁶	64	1000000	40	100
2⁷	128	10000000	80	200
2⁸	256	100000000	100	400
2⁹	512	1000000000	200	1000
2¹⁰	1024	10000000000	400	2000
2¹¹	2048	100000000000	800	4000
2¹²	4096	1000000000000	1000	10000

How to Choose the Right Converter for Your Task

If You Are...	Use This Converter	Why
Learning how computers work	Binary ↔ Decimal	Most fundamental conversion
Web developer designing colours	Decimal ↔ Hexadecimal	RGB to hex colour codes
Programmer debugging memory	Binary ↔ Hexadecimal	Memory addresses in hex
System administrator	Decimal ↔ Octal	chmod file permissions
Working with legacy systems	Binary ↔ Octal	Old Unix/PDP systems
Cross-system debugging	Hexadecimal ↔ Octal	Connect hex and octal worlds

Common Number System Mistakes to Avoid

Mistake #1: Confusing Binary Digits with Decimal Digits

What people do: They treat binary 10 as decimal ten (it is actually two).

Why it is wrong: Binary 10 = 2 decimal, not 10 decimal. In binary, each position doubles, not adds ten.

What to do instead: Always remember: binary digits are powers of 2, not powers of 10.

Mistake #2: Using A-F Characters in Octal or Decimal

What people do: They type "FF" into an octal or decimal converter and wonder why it fails.

Why it is wrong: Octal only uses digits 0-7; decimal only uses 0-9. Letters A-F are only valid in hexadecimal.

What to do instead: Check which number system you are entering before typing letters.

Mistake #3: Forgetting to Group Binary from the Right (Not Left)

What people do: They group binary digits starting from the leftmost digit.

Why it is wrong: The least significant bit (LSB) is on the RIGHT. Always group from the right, adding leading zeros to the leftmost group if needed.

What to do instead: Start grouping from the rightmost digit (position 0).

Mistake #4: Forgetting That chmod Octal Is Three Digits (Owner, Group, Others)

What people do: They use a single octal digit for chmod.

Why it is wrong: chmod requires three digits: one for owner, one for group, one for others. 7 alone is not valid — must be 777, 755, 644, etc.

What to do instead: Always use three digits for chmod permissions (or four in some systems with sticky bits).

Frequently Asked Questions About Number Systems

What is binary and why do computers use it?

Binary (base-2) uses only digits 0 and 1. Computers use binary because transistors have only two reliable states: ON (1) and OFF (0). Every piece of data on your computer — every character in this text, every pixel on your screen, every sound in the music you hear — is ultimately stored as binary. Decimal would require ten-state components, which are less reliable and more expensive.

What is hexadecimal used for in programming?

Hexadecimal (base-16) is used extensively in programming because it is 4× more compact than binary while maintaining an exact correspondence. One hex digit equals 4 binary bits. Specific uses include: HTML/CSS colour codes (#FF5733), memory addresses in debuggers (0x7FFD8A3F2B1C), MAC addresses (00:1A:2B:3C:4D:5E), IPv6 addresses, assembly language, and binary file viewers.

What is octal used for today?

Octal (base-8) is primarily used for Unix/Linux file permissions via the chmod command (e.g., chmod 755). Each octal digit (0-7) represents exactly 3 binary bits, perfectly matching the three permission flags for each user class (read, write, execute). Octal is also used in legacy systems like the PDP-11 and some digital electronics applications.

How do I convert binary to decimal?

Multiply each binary digit by 2 raised to its position (starting from 0 on the right), then sum all results. For example, binary 1010 = (1×8) + (0×4) + (1×2) + (0×1) = 8 + 0 + 2 + 0 = 10 decimal. Use our Binary Decimal Converter for instant results with any binary number.

How do I convert decimal to hexadecimal?

Repeatedly divide the decimal number by 16, recording remainders (0-15, with 10-15 as A-F). Read remainders from bottom to top. For example, 255 decimal ÷ 16 = 15 remainder 15 → FF hex. Use our Decimal Hexadecimal Converter for instant conversion of any decimal number.

How many digits are in each number system?

Binary uses 2 digits: 0 and 1. Decimal uses 10 digits: 0-9. Hexadecimal uses 16 digits: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). Octal uses 8 digits: 0-7. Each system is named for its base (number of digits). The relationship between systems: 1 hex digit = 4 bits, 1 octal digit = 3 bits.

Summary: Master All Number Systems

Here is what you learned today:

• ✅ Binary (base 2) — uses 0 and 1; the fundamental language of computers. Every digital device ultimately operates in binary.

• ✅ Decimal (base 10) — uses 0-9; the everyday number system humans use. Bridging decimal and binary is essential for understanding computing.

• ✅ Hexadecimal (base 16) — uses 0-9 and A-F; 4× more compact than binary. Essential for web colours (#FF5733), memory addresses, and debugging.

• ✅ Octal (base 8) — uses 0-7; primarily used for Unix/Linux file permissions. chmod 755 is octal.

• ✅ 1 hex digit = 4 bits, 1 octal digit = 3 bits — these relationships make conversion straightforward.

• ✅ Use our 6 free converters — instantly convert between any number systems for programming, web design, or studying.

Your Next Step

Stop memorizing conversion tables. Here is what to do right now:

1. Bookmark the Binary Decimal Converter for everyday binary work
2. Use the Decimal Hexadecimal Converter for web colour codes
3. Use the Decimal Octal Converter for chmod permission calculations
4. Reference the conversion table above when studying
5. Share the six converters with colleagues learning number systems

Master binary, decimal, hex, and octal — the languages of computing.

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Disclaimer: This guide is for educational and practical computing purposes. If you are managing sensitive file permissions on production servers, always verify chmod commands before applying. Incorrect permissions can cause security vulnerabilities or system issues.