Binary and Decimal Number System Conversion

Introduction

Every piece of data inside a computer — text, images, audio, instructions — is ultimately represented as a sequence of 0s and 1s. This is the binary (base-2) number system. As humans, we naturally think in decimal (base-10). Knowing how to move between the two systems is a foundational skill in computer science.

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Binary to Decimal

Binary is a positional number system where each digit (bit) represents a power of 2, starting from the rightmost position (position 0).

Method: Positional Value (Powers of 2)

For a binary number $b_n b_{n-1} \ldots b_1 b_0$:

$\text{Decimal} = b_n \times 2^n + \cdots + b_1 \times 2^1 + b_0 \times 2^0$

Example 1 — Convert 1011 to decimal

Position	3	2	1	0
Bit	1	0	1	1
Value	2³=8	2²=4	2¹=2	2⁰=1

$1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 8 + 0 + 2 + 1 = \mathbf{11}$

Example 2 — Convert 11010110 to decimal

Position	7	6	5	4	3	2	1	0
Bit	1	1	0	1	0	1	1	0
Value	128	64	32	16	8	4	2	1

$128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = \mathbf{214}$

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Decimal to Binary

Method: Repeated Division by 2

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Use the quotient as the next number to divide.
4. Repeat until the quotient is 0.
5. Read the remainders from bottom to top — that is your binary result.

Example 1 — Convert 13 to binary

Step	Number	÷ 2	Quotient	Remainder
1	13	÷ 2	6	1
2	6	÷ 2	3	0
3	3	÷ 2	1	1
4	1	÷ 2	0	1

Read remainders bottom-to-top: 1101

Verify: $8 + 4 + 0 + 1 = 13$ ✓

Example 2 — Convert 214 to binary

Step	Number	Quotient	Remainder
1	214	107	0
2	107	53	1
3	53	26	1
4	26	13	0
5	13	6	1
6	6	3	0
7	3	1	1
8	1	0	1

Read bottom-to-top: 11010110

Verify: $128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214$ ✓

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Quick Reference Table (0–15)

Decimal	Binary	Decimal	Binary
0	0000	8	1000
1	0001	9	1001
2	0010	10	1010
3	0011	11	1011
4	0100	12	1100
5	0101	13	1101
6	0110	14	1110
7	0111	15	1111

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Tips & Common Mistakes

Reading direction matters. When you collect remainders during division, always read from the last remainder to the first (bottom-to-top). Reversing this is the most common error.

Leading zeros. In practice, binary numbers are often padded to a fixed width (e.g., 8-bit: 00001101). The leading zeros don't change the value.

Verify your work. After converting decimal → binary, quickly convert back using the positional method to confirm you get the original number.

Powers of 2 cheat sheet:

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

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Summary

Direction	Method	Key Step
Binary → Decimal	Positional value	Multiply each bit by its power of 2, then sum
Decimal → Binary	Repeated division by 2	Collect remainders, read bottom-to-top

Once you are comfortable with base-2 and base-10, expanding to hexadecimal (base-16) — the system used in memory addresses, color codes, and debugging — becomes straightforward.