Binary Calculator
Convert between binary, octal, and decimal number systems.
Binary Calculator
What Is the Binary Calculator?
A binary calculator performs arithmetic and conversions in base two, the number system underlying every modern computer. Binary uses only 0 and 1, mapping directly onto electronic switches that are either off or on, which is why all software ultimately reduces to binary patterns. This tool adds, subtracts, multiplies, and divides binary numbers, converts between binary, decimal, and hexadecimal, and demonstrates carries the way schoolchildren learn them in base ten. Programmers use it to read memory addresses and bit flags, networking students use it for subnet masks, and computer science students use it to understand exactly how hardware represents numbers.
Key Statistics
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2 symbols
Binary uses only 0 and 1, matching the on and off states of transistors
Source: Computer architecture fundamentals
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1679
Year Leibniz published his work on the binary numeral system
Source: History of mathematics
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8 bits
One byte, enough to represent 256 distinct values
Source: Computing convention
The Formula
Decimal to Binary: Repeatedly divide by 2 and read remainders backward. Binary to Decimal: Sum of (digit x 2^position). Octal follows the same pattern with base 8.
Worked Examples
The decimal number 42 (the Answer to the Ultimate Question) converts to binary 101010 and octal 52. The 8-bit binary number 11010110 equals decimal 214, which is one byte in computer memory.
Reading a binary number
- Take 1011 in binary
- Place values run 8, 4, 2, 1 from left to right
- Sum the active positions: 8 plus 0 plus 2 plus 1
Binary 1011 equals decimal 11.
Real World Use Cases
Programming work
Decode bit flags, permissions masks, and memory values that tools display in hex or binary.
Networking study
Work subnet masks and IP ranges, which are binary arithmetic in disguise.
Computer science coursework
Practice number system conversions that exams and interviews expect.
Expert Tips
- ✓ Group binary digits in fours from the right; each group maps to exactly one hexadecimal digit.
- ✓ Learn powers of two through 1024; they appear constantly in memory sizes and conversions.
- ✓ Binary multiplication is trivially shift and add, which is how hardware multiplies at all.
- ✓ Two's complement representation makes subtraction work with addition circuits; negative numbers wrap around high bits.
Frequently Asked Questions
How do I convert binary to decimal? ▼
Multiply each binary digit by 2 raised to its position, counting from the right starting at zero, then add the results. Binary 1010 is 1 times 8 plus 1 times 2, which equals 10 in decimal.
How do I convert decimal to binary? ▼
Divide the number by 2 repeatedly and record the remainders, then read them in reverse order. Decimal 13 divides into binary 1101. This calculator does the conversion instantly.
How do I add binary numbers? ▼
Add bit by bit from right to left, carrying a 1 when a column sums to 2. In binary, 1 plus 1 equals 10, so you write 0 and carry 1, just like carrying in decimal addition.
Why does a computer use binary? ▼
Computers use binary because their hardware works with two states, like a switch that is on or off, or voltage that is high or low. Representing everything as 1s and 0s makes that hardware simple and reliable.
How do I convert binary to hexadecimal? ▼
Group the binary digits into sets of four from the right, then convert each group. Binary 1111 is F in hex, and 1010 is A, so 11111010 is FA. Four bits always map to one hex digit.
How do I count in binary? ▼
Counting in binary goes 0, 1, 10, 11, 100, 101, 110, 111, 1000, and so on. Once a digit reaches 1, the next step rolls it over and increments the next place, like 9 rolling to 10 in decimal.
How do I convert decimal to binary? ▼
Divide repeatedly by two, writing each remainder; reading remainders bottom to top gives the binary form.
Why do computers use binary? ▼
Two state devices are the most reliable electronics to build, and binary logic maps directly onto boolean algebra.
Common Mistakes to Avoid
- ⚠ Entering digits other than 0 or 1 in the binary field. Typing 2, spaces, or letters produces no conversion. Binary only uses 0 and 1, and the calculator rejects invalid characters.
- ⚠ Misreading binary place values when cross-checking results. In binary, the rightmost digit is the 1s place (2^0), not the leftmost. The binary number 1000 equals 8, not 1000, and a string of 8 ones (11111111) equals 255, not 99,999,999.
- ⚠ Confusing octal output with decimal. The octal number 10 equals 8 in decimal, not 10. When using the octal field, remember that digits 8 and 9 are invalid in base 8.
- ⚠ Mixing up place value direction. Binary places grow leftward by doubling, so 10 is two but 10000000 is 128.
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