Binary Mathematics: The Foundation of Digital Computing, Converting and Operations
Published: 2023-09-12 17:48:04

Binary math is a system of mathematics that uses only two digits: 0 and 1. It's the foundation of digital computing and is used to represent and manipulate information in computers. In binary math:

Numbers: Numbers are represented using combinations of 0s and 1s. Each digit in a binary number represents a power of 2, starting from the right and increasing by one for each digit to the left. For example, the binary number "1101" represents 1*(2^3) + 1*(2^2) + 0*(2^1) + 1*(2^0), which equals 13 in decimal (base-10) notation.

Operations: Basic arithmetic operations like addition, subtraction, multiplication, and division can be performed using binary digits. These operations follow similar rules to those in decimal math but with simpler outcomes because there are only two digits involved.

Logic: Binary math is closely tied to binary logic, where 0 typically represents "false" and 1 represents "true." This logic is fundamental in computer programming, where it's used to make decisions and perform computations.

Data Storage: Computers use binary math to store and process data. All data, including text, images, and videos, is ultimately represented in binary form as combinations of 0s and 1s.


Operations:

- Binary Addition:

Binary addition is similar to decimal addition but with only two possible values: 0 and 1.
When adding two binary numbers, you start from the rightmost bit and move left.
If you add 0 and 0, the result is 0. If you add 1 and 0, the result is 1. If you add 1 and 1, the result is 10 (carry-over 1 to the next bit).
Carry-overs are important in binary addition, just as in decimal addition. They propagate from right to left until there's no more carry-over.
Example: 1101 + 101 = 10010 (in binary).


- Binary Subtraction:

Binary subtraction also resembles decimal subtraction but with two possible values: 0 and 1.
When subtracting binary numbers, you start from the rightmost bit and move left.
If you subtract 0 from 0, the result is 0. If you subtract 1 from 0, you borrow 1 from the next higher bit (if possible) and subtract 1 from 10, resulting in 1. If you subtract 1 from 1, the result is 0.
Borrowing is crucial in binary subtraction, similar to decimal subtraction.
Example: 1101 - 101 = 100 (in binary).


- Binary Multiplication:

Binary multiplication involves multiplying two binary numbers, bit by bit.
Start with the rightmost bit of the second number and multiply it by each bit of the first number, shifting one position to the left with each step.
Just like in decimal multiplication, you can have carry-overs when multiplying. If the product of two bits is 2, carry over 1.
Example: 1011 * 10 = 10110 (in binary).


- Binary Division:

Binary division is similar to decimal long division but uses 0 and 1 as digits.
You start by comparing the leftmost bits of the dividend and the divisor.
If the divisor is smaller or equal to the dividend, you subtract and write down the quotient (0 or 1) and bring down the next bit.
Continue this process until you've worked through all the bits.
Example: 10101 / 11 = 11 with a remainder of 1 (in binary).




- Conversion to Base-8 (Octal):

1) Group the binary digits: 001 101 101.
2) Convert each group to octal: 1 5 5.
3) Combine the octal digits: 155.

So, 1101101 in base-2 is equivalent to 155 in base-8.


- Conversion to Base-10 (Decimal):

1) Start from the rightmost digit.
2) 1 * 2^0 + 0 * 2^1 + 1 * 2^2 + 1 * 2^3 + 0 * 2^4 + 1 * 2^5 + 1 * 2^6 = 1 + 0 + 4 + 8 + 0 + 32 + 64 = 109.