Flip a single switch—off to on—and you’ve captured the essence of binary. From smartphones to spacecraft, the modern digital world is built on numbers that use just two symbols: 0 and 1. A Binary Calculator turns that elegant simplicity into speed, letting you convert values, add, subtract, multiply, and divide in base‑2 with clarity and confidence.

What is the binary system?

The binary system is a base‑2 numeral system that uses only two digits—0 and 1—where each digit is called a bit. Like the decimal system (base‑10), binary is positional: each place represents a power of its base, so binary place values progress as 20,21,22,23,20,21,22,23, and so on. This makes binary ideal for digital circuits that detect two states such as on/off or true/false.

Why computers use binary

Digital hardware prefers certainty. Detecting two stable states is far more robust than distinguishing among ten, so binary maps cleanly onto logic gates and memory cells. This practicality is why almost all computing—data, images, audio—ultimately reduces to patterns of bits.

Quick binary–decimal conversions

• Place-value rule: sum the powers of two wherever there is a 1. Example: 101112=1×24+0×23+1×22+1×21+1×20=16+4+2+1=2310101112=1×24+0×23+1×22+1×21+1×20=16+4+2+1=2310.
• Common pairs:
  • 0 ↔ 0
  • 1 ↔ 1
  • 2 ↔ 10
  • 3 ↔ 11
  • 4 ↔ 100
  • 7 ↔ 111
  • 8 ↔ 1000
  • 10 ↔ 1010
  • 16 ↔ 10000
  • 20 ↔ 10100

Decimal to binary: step‑by‑step

To convert a decimal number:

1. Find the largest power of 2 less than or equal to the number.
2. Subtract it and repeat with the remainder.
3. Mark a 1 for each used power of two and 0 otherwise.

Example with 18: powers are 24=1624=16 and 21=221=2. So 18=16+218=16+2 → binary is 10010.

Binary to decimal: step‑by‑step

Identify all positions that have a 1 and add the corresponding powers of two.
Example: 100102=1×24+1×21=16+2=18100102=1×24+1×21=16+2=18.

Binary addition

Binary addition mirrors decimal addition but carries at 2 instead of 10:

• 0+0=0
• 0+1=1
• 1+0=1
• 1+1=0 with a carry of 1 (i.e., 10 in binary)

Tip: Watch for a carry‑in from the right. For example, if a prior column carried 1, then 1 (carry) + 1 + 1 = 1 with a carry (since 3 in decimal is 11 in binary).

Binary subtraction

Borrowing occurs only for 1−from−0 cases. When borrowing, the 0 becomes “2” in binary terms, so 2−1=12−1=1, and the bit you borrow from is reduced by 1. If the next column is also 0, keep borrowing left until a 1 is found.

Rules recap:

• 0−0=0
• 1−0=1
• 1−1=0
• 0−1=1 with a borrow from the next higher bit

Binary multiplication

Because digits are only 0 or 1, partial products are either 0 or the multiplicand:

• 0×x=0
• 1×x=x

Shift left one position for each move to the next bit (just like appending zeros in decimal). Then add the partial products in binary.

Example outline:

• Multiply the multiplicand by each bit of the multiplier.
• Shift left for each next bit position.
• Add all shifted rows using binary addition.

Binary division

Binary long division mirrors decimal long division:

• Compare the divisor to the current portion of the dividend.
• If the divisor “fits,” write 1 in the quotient and subtract; if not, write 0 and bring down the next bit.
• Continue until all bits are processed or a remainder remains.

How a Binary Calculator helps

A Binary Calculator streamlines everything:

• Instant conversions between binary and decimal.
• Fast binary addition, subtraction, multiplication, and division with correct carries, borrows, and shifts.
• Clear, step‑wise workings to learn the logic while computing.

Practical tips for clean results

• Group bits in fours to improve readability: 10010 → 1 0010.
• Validate with a quick sanity check by converting final binary back to decimal.
• For arithmetic, write out carries and borrows explicitly to avoid off‑by‑one errors.

Key patterns to remember

• Each shift left in binary multiplies by 2; each shift right divides by 2 (discarding fractions).
• The highest 1‑bit marks the largest power of two in the number.
• Consecutive trailing zeros indicate factors of 2.