Question 1

How do you convert binary to decimal by hand?

Accepted Answer

Give each digit a place value that doubles as you move left — 1, 2, 4, 8, 16, 32, and so on (each one is 2 raised to that digit's position, counting from 0 on the right). Multiply every binary digit by its place value and add up only the results where the digit is 1. For 1101, that's 1×8 + 1×4 + 0×2 + 1×1 = 8 + 4 + 0 + 1 = 13. The reverse — decimal to binary — works by repeatedly dividing by 2 and reading the remainders from bottom to top, which is exactly what the Dec → Bin mode above automates for you.

Question 2

How does binary addition handle carrying?

Accepted Answer

Binary addition follows the same carry logic as decimal, just with only two digits instead of ten. Adding two 1s gives 10 in binary (0 with a carry of 1), so 1+1=10, 1+1+1(carry)=11, and so on. Working through 1010 + 0110 column by column from the right: 0+0=0, 1+1=10 (write 0, carry 1), 0+1+1(carry)=10 (write 0, carry 1), 1+0+1(carry)=10 (write 0, carry 1), then the final carry drops down — giving 10000, which checks out since 10(decimal) + 6(decimal) = 16(decimal) = 10000(binary).

Question 3

What is two's complement and why is it used for negative numbers?

Accepted Answer

Two's complement is the standard way computers represent negative integers in binary without needing a separate minus sign or subtraction circuit. To negate a number, you flip every bit and add 1 — for example, in 8 bits, 5 is 00000101, so -5 becomes 11111011. Its advantage is that addition, subtraction, and comparison all work with the same hardware circuitry regardless of sign, and there's exactly one representation of zero (unlike the sign-and-magnitude scheme this calculator uses for readability, which keeps a separate sign and shows the magnitude's binary directly).

Question 4

Why do computers use binary instead of decimal?

Accepted Answer

Computer memory and logic circuits are built from transistors that are cleanest to design as two-state switches — on or off, high voltage or low voltage — which maps naturally onto just two digits, 0 and 1. Building reliable circuitry for ten distinct voltage levels (to represent decimal digits directly) would be far more error-prone and expensive than distinguishing just two states. Binary also pairs cleanly with hexadecimal and octal, which is why hex is used as a compact shorthand for binary in programming — see our hex calculator to convert between the two.

Dec	Binary	Hex	Oct	Dec	Binary	Hex	Oct
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	8	10	9	01001	9	11
10	01010	A	12	11	01011	B	13
12	01100	C	14	13	01101	D	15
14	01110	E	16	15	01111	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

Bit	Position (2ⁿ)	Place Value	Contributes
1	2³	8	1 × 8 = 8
1	2²	4	1 × 4 = 4
0	2¹	2	0 × 2 = 0
1	2⁰	1	1 × 1 = 1

Binary
Calculator

Binary Arithmetic

📋 Binary Reference Table (0–31)

How Binary Numbers Work

Place Value Example — Binary 1101

Conversion Method

Frequently Asked Questions

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