Binary System Cheatsheet

Decimal to Binary

To convert a decimal number to binary, divide the decimal number by 2 successively and take the remainders in reverse order.

Decimal Number: 10

10 / 2 = 5 (remainder 0)
5 / 2 = 2 (remainder 1)
2 / 2 = 1 (remainder 0)
1 / 2 = 0 (remainder 1)

Binary Number: 1010

Binary to Decimal

To convert a binary number to decimal, multiply the binary digits by the corresponding powers of 2 and sum the results.

Binary Number: 1010

1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 8 + 0 + 2 + 0 = 10

Decimal Number: 10

Bits and Bytes

A bit is the smallest unit of information in a binary system, and it can have the value of 0 or 1. A byte is composed of 8 bits and is the basic unit of storage in most computer systems.

Bit: 1
Byte: 01011011

Representation of Positive Integers

Integers are represented using a fixed number of bits (usually 32 or 64 bits) in binary format.

Integer: 10101110
Real: 01000000101011000000000000000000

Representation of Negative Integers

Two’s complement is used.

1. Invert all bits.
2. Add 1 to the result.

Original Number: 1010
One's Complement: 0101
Two's Complement: 0101 + 1 = 0110

Representation of Fractional Numbers

Real numbers are represented by the IEEE 754 standard, which uses a combination of bits to represent the sign, exponent, and mantissa of the number.

Binary Addition

1. Add the digits starting from the right.
2. Carry over if there is a carry.

  1010
+ 0110
------
 10000

Binary Subtraction

1. Subtract the digits starting from the right.
2. Borrow if necessary.

  1010
- 0110
------
  0100

Binary Multiplication

Binary multiplication is performed similarly to decimal multiplication, but only multiplications by 0 and 1 are used.

  1010
* 0011
------
 10100
1010
------
11110

Binary AND

// AND operation
1010 & 1100 = 1000

Binary OR

// OR operation
1010 | 1100 =  1110

Binary XOR

// XOR operation
1010 ^ 1100 = 0110

Bit Shift

// Left shift
1011 << 1 = 10110

# Right shift
1011 >> 1 = 101