About the Binary Converter
The binary converter is a handy tool that will enable you to perform a conversion of numbers quickly. You will be able to use it to convert decimal numbers to binary representation, including support for negative numbers using one's and two's complement notation.
What is the Binary System?
We usually operate in the decimal system. That means that we use ten different digits, from 0 to 9. If you write down a number, for example, 345, each consecutive digit of this number corresponds to a different power of ten. That means that:
- Digit 5 corresponds to 10⁰ (5 is equal to 5×100)
- Digit 4 corresponds to 10¹ (40 is equal to 4×101)
- Digit 3 corresponds to 10² (300 is equal to 3×102)
In the binary system, there are only two available digits: 0 and 1. That means that the digits of every number correspond to powers of 2 instead of corresponding to powers of 10.
Converting from Decimal to Binary
You can use a very simple and efficient algorithm to convert numbers from the decimal to binary system:
- Take your initial number. Divide it by 2.
- Note down the remainder. It is going to be equal to 0 or 1. This will be the last digit of the binary number (the rightmost one).
- Take the quotient. It is your new "initial number".
- Keep repeating the above steps, each time adding the remainder to the left of previously obtained digits.
Example: Converting 19 to binary
- 19/2 = 9, remainder 1
- 9/2 = 4, remainder 1
- 4/2 = 2, remainder 0
- 2/2 = 1, remainder 0
- 1/2 = 0, remainder 1
Reading from bottom to top: 19 = 10011 in binary
Converting from Binary to Decimal
For example, for the binary number 110011, we would have the following steps:
- 1×2 = 2
- (2+1)×2 = 6
- (6+0)×2 = 12
- (12+0)×2 = 24
- (24+1)×2 = 50
- 50+1 = 51
110011 = 51 in decimal
Converting Negative Numbers
We are used to simply adding a minus symbol in front of the number if we want to express a negative number in the decimal system. But the binary system does not allow the minus symbol. So how can we represent negative numbers in binary?
The general concept to express negative numbers in the binary system is the signed notation. That means that the first bit indicates the sign of the number: 0 means positive, 1 is a negative value. The signed notation has two representations:
One's Complement
The one's complement of a negative number in binary is achieved by switching all digits of the opposite positive number to opposite bit values.
Two's Complement
The two's complement of a negative number in binary is achieved by switching all digits of the opposite positive number to opposite bit values and adding 1 to the number.
Example: Converting -87 to Binary (8-bit)
- Find the binary representation of the positive number 87:
0101 0111 - Switch all digits to the opposite:
1010 1000(One's complement) - Add 1 to the one's complement:
1010 1001(Two's complement)
How to Use the Binary Converter?
- Choose the number of bits: 8, 16, 32, or 64 bits. More bits allow for a larger range of numbers.
- Enter your decimal value: Can be positive or negative. The calculator displays the range for your selected bit size.
- Click Calculate: The calculator will show:
- Binary representation of the positive value
- One's complement (for negative numbers)
- Two's complement (for negative numbers)
- Hexadecimal and octal representations
Bit Size Ranges
| Bits | Minimum Value | Maximum Value | Common Use |
|---|---|---|---|
| 8-bit | -128 | 127 | Byte, small integers |
| 16-bit | -32,768 | 32,767 | Short integers, Unicode characters |
| 32-bit | -2,147,483,648 | 2,147,483,647 | Standard integers, IPv4 addresses |
| 64-bit | -9,223,372,036,854,775,808 | 9,223,372,036,854,775,807 | Long integers, timestamps |
Real-World Applications
- Computer Programming: Understanding how computers store negative numbers
- Digital Electronics: Designing arithmetic circuits
- Binary Subtraction: Using two's complement for efficient subtraction
- Data Types: Choosing appropriate integer types in programming
- Debugging: Analyzing memory dumps and binary data