Binary Calculator

Add column by column from right to left, carrying over when the sum exceeds 1:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Example: 1011 + 1101 → Start from right: 1+1=10 (write 0, carry 1), 1+0+1=10 (write 0, carry 1), 0+1+1=10 (write 0, carry 1), 1+1+1=11. Result: 11000 (decimal 24)

Subtract column by column. When 0 minus 1, borrow from the left (borrowing adds 2 to your current bit):

• 0 − 0 = 0
• 1 − 0 = 1
• 1 − 1 = 0
• 0 − 1 = 1 (borrow 1 from left)

Example: 1100 − 0101 → Rightmost: 0−1 borrow, get 10−1=1. Next: 0−0=0 (reduced by borrow to 1−0−1=0). Result: 0111 (decimal 7)

Works like long multiplication in decimal. Multiply each bit, then shift and add the partial products:

• 0 × anything = 0
• 1 × anything = that number

Example: 101 × 11 → 101×1 = 101, 101×1 shifted left = 1010. Add: 101 + 1010 = 1111 (decimal 15, i.e. 5×3)

Uses long division — check if the divisor fits into the current portion, subtract if yes (write 1), skip if no (write 0):

• If portion ≥ divisor → write 1, subtract
• If portion < divisor → write 0, bring down

Example: 1010 ÷ 10 → 10 fits into 10 once (1), remainder 0, bring down 1 → 01, doesn't fit (0), bring down 0 → 10, fits once (1). Result: 101 (decimal 5, i.e. 10÷2)

1 & 1 = 1

Only outputs 1 when both bits are 1. Everything else gives 0.

Example: 1010 & 1100 = 1000

Use: Masking bits, checking flags

0 | 1 = 1

Outputs 1 when at least one bit is 1. Only 0|0 gives 0.

Example: 1010 | 1100 = 1110

Use: Setting flags, combining values

1 ^ 0 = 1

Outputs 1 when the bits are different. Same bits give 0.

Example: 1010 ^ 1100 = 0110

Use: Toggling bits, encryption, checksums

~1 = 0

Inverts each bit: every 1 becomes 0 and every 0 becomes 1.

Example (4-bit): ~1010 = 0101

Use: One's complement, bit inversion

bits << n

Shifts all bits left by n positions. Zeros fill from the right. Each shift doubles the value.

Example: 0011 << 2 = 1100 (3→12)

Use: Fast multiplication by powers of 2

bits >> n

Shifts all bits right by n positions. Zeros fill from the left. Each shift halves the value.

Example: 1100 >> 2 = 0011 (12→3)

Use: Fast division by powers of 2

Sum of (Bit × 2^n)

Starting from the right, each bit position has a value (1, 2, 4, 8, 16...). Multiply each bit by its position value and add them up:

1101 → (1×8) + (1×4) + (0×2) + (1×1) = 8+4+0+1 = 13

Divide by 2, read remainders

Keep dividing by 2 and write down the remainder. Read the remainders from bottom to top:

13 → 13÷2=6 R1, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1 → 1101

Group 4 bits → 1 hex digit

Split the binary number into groups of 4 bits from right to left. Convert each group: 0000=0, 1001=9, 1010=A, 1111=F.

10101111 → 1010 | 1111 → A | F = AF

Group 3 bits → 1 octal digit

Split the binary number into groups of 3 bits from right to left. Convert each group: 000=0, 111=7.

001010 → 001 | 010 → 1 | 2 = 12 (octal)