Binary Calculator — How to Convert Binary to Decimal and Perform Binary Arithmetic

Binary is the number system that powers every computer, smartphone, and digital device on Earth. While humans naturally use the decimal (base-10) system, computers work with base-2 — two digits, 0 and 1, corresponding to the off and on states of electronic transistors. Understanding binary is fundamental to computer science, digital electronics, and programming.

What Is Binary (Base-2)?

The binary number system uses only two digits: 0 and 1. Each digit in a binary number is called a bit (binary digit). Groups of bits form larger units:

Unit	Bits	Max value
Bit	1	1
Nibble	4	15
Byte	8	255
Word	16	65,535
Double Word	32	~4.3 billion

How to Convert Binary to Decimal Step by Step

Each position in a binary number represents a power of 2, starting at 2⁰ = 1 on the right and doubling with each position to the left.

Positional values:

```

Position: 7 6 5 4 3 2 1 0

Value: 128 64 32 16 8 4 2 1

```

Example: Convert 1011₂ to decimal

1. Write the binary digits with their positional values: 1×8, 0×4, 1×2, 1×1

2. Multiply: 8 + 0 + 2 + 1 = 11₁₀

Example: Convert 11010110₂ to decimal

• 1×128 + 1×64 + 0×32 + 1×16 + 0×8 + 1×4 + 1×2 + 0×1
• = 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214₁₀

How to Convert Decimal to Binary

The method is repeated division by 2 — divide the number by 2, record the remainder, and continue dividing the quotient until you reach 0. Read the remainders from bottom to top.

Example: Convert 45₁₀ to binary

Division	Quotient	Remainder
45 ÷ 2	22	**1**
22 ÷ 2	11	**0**
11 ÷ 2	5	**1**
5 ÷ 2	2	**1**
2 ÷ 2	1	**0**
1 ÷ 2	0	**1**

Read remainders bottom to top: 101101₂

Verify: 32 + 0 + 8 + 4 + 0 + 1 = 45 ✓

Binary Addition

Binary addition uses the same column-by-column approach as decimal, but with only two digits:

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Example: 1011₂ + 0111₂

```

1011

+ 0111

------

10010

```

= 10010₂ = 18₁₀ (11 + 7 = 18 ✓)

Binary Subtraction

Binary subtraction works by borrowing from the next column when subtracting 1 from 0 (just as you borrow 10 when subtracting a larger decimal digit from a smaller one).

Example: 1100₂ − 0101₂

• 1100 = 12₁₀, 0101 = 5₁₀
• 12 − 5 = 7 = 0111₂

Binary Multiplication

Binary multiplication is actually simpler than decimal because you only ever multiply by 0 or 1:

• Any number × 0 = 0
• Any number × 1 = that number

Then add the partial products using binary addition.

Example: 101₂ × 11₂ (5 × 3 = 15)

```

101

× 11

------

101 (101 × 1)

1010 (101 × 1, shifted left)

------

1111

```

= 1111₂ = 15₁₀ ✓

Why Computers Use Binary

Computers use binary because transistors — the fundamental building blocks of processors — have two stable states:

• 0 = low voltage (off)
• 1 = high voltage (on)

This binary physical property maps perfectly to binary arithmetic. Storing or transmitting a value with just two states is extremely reliable — a transistor is either on or off, unlike analogue systems where signal degradation is a problem.

Modern processors have billions of transistors, each performing binary operations billions of times per second.

Binary in Computer Science

IP Addresses: IPv4 addresses are 32 bits. 192.168.1.1 in binary:

• 192 = 11000000, 168 = 10101000, 1 = 00000001, 1 = 00000001

ASCII: Text characters are encoded as binary numbers. The letter 'A' is 65₁₀ = 01000001₂.

Colours: RGB colour values are each 8-bit numbers (0–255). The colour white = 11111111 11111111 11111111₂.

Related Resources

Related Calculators

• Hex Calculator — Convert binary numbers to hexadecimal.
• Percentage Calculator — Compute fractional values.

External Authority Resources

• Wikipedia: Binary Number System — Thorough history and operations of base-2.
• Wolfram MathWorld: Binary — Technical details of binary notation.