## Binary Operation (Digital Electronics)

Binary Numbers

**In mathematics and computer science , the Binary numeral system, or base-2 numeral system, represents numeric values using two symbols: 0 and 1. More specifically, the usual base2 system is a positional system with a radix of 2. Numbers represented in this system are commonly called binary numbers. Because of its straightforward implementation in digital elecronic circuitry using logic gates, the Binary system is used internally by almost all modern computer and computer based devices such as mobile phones.**

**Binary Operations**

BINARY ADDITION

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0, and carry 1 to the next more significant bit

For example,

00011010 + 00001100 = 00100110 1 1 carries

0 0 0 1 1 0 1 0 = 26(base 10)

+ 0 0 0 0 1 1 0 0 = 12(base 10)

= 0 0 1 0 0 1 1 0 = 38(base 10)

00010011 + 00111110 = 01010001 1 1 1 1 1 carries

0 0 0 1 0 0 1 1 = 19(base 10)

+ 0 0 1 1 1 1 1 0 = 62(base 10)

=0 1 0 1 0 0 0 1 = 81(base 10)

Note: The rules of binary addition (without carries) are the same as the truths of the XOR gate.

Rules of Binary Subtraction

0 – 0 = 0

0 – 1 = 1, and borrow 1 from the next more significant bit

1 – 0 = 1

1 – 1 = 0

For example,

00100101 – 00010001 = 00010100 0 borrows

0 0 1 10 0 1 0 1 = 37(base 10)

– 0 0 0 1 0 0 0 1 = 17(base 10)

= 0 0 0 1 0 1 0 0 = 20(base 10)

00110011 – 00010110 = 00011101 0 10 1 borrows

0 0 1 1 0 10 1 1 = 51(base 10)

– 0 0 0 1 0 1 1 0 = 22(base 10)

= 0 0 0 1 1 1 0 1 = 29(base 10)

Rules of Binary Multiplication

0 x 0 = 0

0 x 1 = 0

1 x 0 = 0

1 x 1 = 1, and no carry or borrow bits

For example,

00101001 × 00000110 = 11110110 0 0 1 0 1 0 0 1 = 41(base 10)

× 0 0 0 0 0 1 1 0= 6(base 10)

0 0 0 0 0 0 0 0

0 0 1 0 1 0 0 1

0 0 1 0 1 0 0 1

0 0 1 1 1 1 0 1 1 0 = 246(base 10)

00010111 × 00000011 = 01000101 0 0 0 1 0 1 1 1 = 23(base 10)

× 0 0 0 0 0 0 1 1 = 3(base 10)

1 1 1 1 1 carries

0 0 0 1 0 1 1 1

0 0 0 1 0 1 1 1

0 0 1 0 0 0 1 0 1 = 69(base 10)

Note: The rules of binary multiplication are the same as the truths of the AND gate.

Another Method: Binary multiplication is the same as repeated binary addition; add the multicand to itself the multiplier number of times.

For example,

00001000 × 00000011 = 00011000 1 carries

0 0 0 0 1 0 0 0 = 8(base 10)

0 0 0 0 1 0 0 0 = 8(base 10)

+ 0 0 0 0 1 0 0 0 = 8(base 10)

0 0 0 1 1 0 0 0 = 24(base 10)

Notes

Binary Number System

System Digits: 0 and 1

Bit (short for binary digit): A single binary digit

LSB (least significant bit): The rightmost bit

MSB (most significant bit): The leftmost bit

Upper Byte (or nybble): The right-hand byte (or nybble) of a pair

Lower Byte (or nybble): The left-hand byte (or nybble) of a pair

Binary Equivalents

1 Nybble (or nibble) = 4 bits

1 Byte = 2 nybbles = 8 bits

1 Kilobyte (KB) = 1024 bytes

1 Megabyte (MB) = 1024 kilobytes = 1,048,576 bytes

1 Gigabyte (GB) = 1024 megabytes = 1,073,741,824 bytes

**Question**

1.Define binary logic?

Binary logic consists of binary variables and logical operations. The variables are designated by the alphabets such as A, B, C, x, y, z, etc., with each variable having only two distinct values: 1 and 0. There are three basic logic operations: AND, OR, and NOT.

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