🎓 Prepared by students from Boğaziçi University

What are Binary Numbers?

Binary is the base-2 number system that uses only two digits, 0 and 1, to represent every value — it's the language computers use internally because each digit maps directly to an electrical on/off state. Every binary number can be converted to the familiar base-10 (decimal) system using place values that are powers of two.

Short answer

A binary number represents a value using only the digits 0 and 1, where each position stands for a power of 2 (1, 2, 4, 8, 16...), and computers use it because a digit's two states map directly to electronic on/off signals.

Binary vs. Decimal

Binary (Base 2)

•Uses only two digits: 0 and 1
•Place values are powers of 2 (1, 2, 4, 8...)
•Native language of computer hardware

Decimal (Base 10)

•Uses ten digits: 0 through 9
•Place values are powers of 10 (1, 10, 100...)
•The number system humans use every day

Try it: interactive calculator

Decimal value (N)

= 1*2^3+0*2^2+1*2^1+0*2^0

Bit 3 (value 8)1 Bit 2 (value 4)0 Bit 1 (value 2)1 Bit 0 (value 1)0

Step-by-step worked examples

Convert binary 1011 to decimal.

Digits from left: d3=1, d2=0, d1=1, d0=1
N = 1·2³ + 0·2² + 1·2¹ + 1·2⁰
N = 8 + 0 + 2 + 1
N = 11

Convert binary 1101 to decimal.

Digits: d3=1, d2=1, d1=0, d0=1
N = 1·8 + 1·4 + 0·2 + 1·1
N = 8 + 4 + 0 + 1
N = 13

Convert decimal 9 to binary using powers of 2.

Largest power of 2 ≤ 9 is 8 (2³): 9 − 8 = 1, so bit3 = 1
Next, 2² = 4 doesn't fit into 1: bit2 = 0
Next, 2¹ = 2 doesn't fit into 1: bit1 = 0
Last, 2⁰ = 1 fits exactly: bit0 = 1 → binary 1001

Flashcards

Quick quiz

Q1.What is binary 1010 in decimal?

Correct answer: B. 1·8 + 0·4 + 1·2 + 0·1 = 10.

Q2.Which digits are allowed in binary?

Correct answer: B. Binary is base-2, so it only uses the digits 0 and 1.

Q3.What decimal value does the leftmost bit represent in the 4-bit number 1000?

Correct answer: D. The leftmost bit in a 4-bit number is the 2³ place, worth 8.

Q4.Why is binary the language of computer hardware?

Correct answer: B. Transistors are naturally two-state (on/off), matching binary's two digits perfectly.

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Common mistakes

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Reading binary digits as if they were decimal, e.g. thinking 10 means ten. — Correct: Binary 10 equals decimal 2 — always convert using powers of 2, not decimal place value.

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Forgetting that the rightmost bit is the least significant (2⁰), not the leftmost. — Correct: Place values grow from right to left: 2⁰, 2¹, 2², 2³...

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Assuming binary can only represent whole numbers. — Correct: Binary can represent fractions too, using negative powers of 2 after a binary point.

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Mixing up bit and byte. — Correct: A bit is a single 0 or 1; a byte is a group of 8 bits (256 possible values).

FAQ

What are binary numbers?

Binary numbers are values written in base-2, using only the digits 0 and 1, where each position is a power of 2.

What is the binary number formula?

N = d₃·2³ + d₂·2² + d₁·2¹ + d₀·2⁰ — sum each digit times its place value (a power of 2).

What are examples of binary numbers?

1011 (=11 in decimal), 1101 (=13), and 1001 (=9) are all valid binary numbers.

How do you calculate a binary number's decimal value?

Multiply each binary digit by its place value (1, 2, 4, 8, 16...) and add the results together.