What is the Binary Calculator?
The binary system is a numerical system that functions virtually identically to the decimal number system that most people are more familiar with. While the standard decimal number system uses the number 10 as its base, the binary system uses a base of 2. Furthermore, although the decimal system uses the digits 0 through 9, the binary system uses only the digits 0 and 1. Each digit in the binary system is referred to as a "bit."
Apart from these fundamental differences, operations such as addition, subtraction, multiplication, and division are all computed following the exact same logical rules as the decimal system.
Almost all modern technology and computers use the binary system due to its ease of implementation in digital circuitry using logic gates. It is much simpler to design hardware that only needs to detect two states: on and off (or true/false, present/absent, high/low voltage). Attempting to use a decimal system in computing would require complex hardware capable of detecting 10 distinct states for the digits 0 through 9.
Typical Conversions
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 7 | 111 |
| 8 | 1000 |
| 10 | 1010 |
| 16 | 10000 |
| 20 | 10100 |
How to Convert Between Decimal and Binary
While working with binary may initially seem confusing, understanding that each binary place value represents 2n, just as each decimal place represents 10n, should help clarify the concept. Take the number 8, for example. In the decimal number system, 8 is positioned in the first decimal place left of the decimal point, signifying the 100 place. Essentially, this means: 8 × 100 = 8 × 1 = 8.
In binary, 8 is represented as 1000. Reading from right to left, the first 0 represents 20, the second 21, the third 22, and the fourth represents 23. Since 23 = 8, a 1 is entered in its position yielding 1000.
10010 = (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (0 × 20)
10010 = 16 + 0 + 0 + 2 + 0 = 18
The step-by-step process to convert from the decimal to the binary system is:
- Find the largest power of 2 that lies within the given number.
- Subtract that value from the given number.
- Find the largest power of 2 within the remainder found in step 2.
- Repeat until there is no remainder.
- Enter a 1 for each binary place value that was found, and a 0 for the rest.
Binary Arithmetic
Binary Addition
Binary addition follows the same rules as addition in the decimal system, except that rather than carrying a 1 over when the values added equal 10, carry-over occurs when the result of addition equals 2.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (carry over the 1, becoming 10)
Binary Subtraction
Borrowing occurs in any instance where the number that is subtracted is larger than the number it is being subtracted from. In binary subtraction, the only case where borrowing is necessary is when 1 is subtracted from 0. When this occurs, the 0 in the borrowing column essentially becomes a "2" (since it borrows from the next power of 2).
- 0 - 0 = 0
- 0 - 1 = 1 (borrow 1, resulting in -1 carried over to the next column)
- 1 - 0 = 1
- 1 - 1 = 0
Binary Multiplication
Binary multiplication is arguably simpler than its decimal counterpart. Since the only values used are 0 and 1, the results that must be added are either exactly the same as the first term, or 0. Just like in decimal multiplication, placeholder 0s need to be added for each subsequent row before doing the final addition.
- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 × 1 = 1
Binary Division
The process of binary division is very similar to long division in the decimal system. The dividend is still divided by the divisor in the same manner, with the only significant difference being the use of binary rather than decimal subtraction. A good understanding of binary subtraction is required for conducting binary division accurately.
Frequently Asked Questions
Computers operate using microswitches and transistors, which can only exist in two distinct states: on (voltage present) or off (no voltage). Binary perfectly maps to this physical limitation, assigning 1 to "on" and 0 to "off", making it highly reliable and extremely fast to process electronically.
A "bit" (short for binary digit) is the smallest unit of data in a computer and has a single binary value, either 0 or 1. A "byte" is a sequence of 8 bits processed as a single unit of information. Because a byte has 8 bits, it can represent 256 different values (28 = 256).
While you can subtract using the borrowing method, computers often use the "Two's Complement" method. You invert all the bits of the number you want to subtract (change 1s to 0s and 0s to 1s), add 1 to that result, and then add it to your original number. This turns complex subtraction into simple addition.
Yes! The logic is identical to standard decimal long multiplication. Because you only multiply by 0 or 1, the intermediate steps are simply writing down either zeroes or a copy of the top number shifted to the left, followed by binary addition to get the final product.
Yes, highly related. Hexadecimal is base-16. Because 16 is a power of 2 (24), every single hexadecimal digit maps exactly to a 4-bit binary sequence (a "nibble"). Programmers use hex to write long, hard-to-read binary strings in a much shorter and more readable format.