1.2. Binary Number System

The Binary Number System is the base-2 number system that represents numeric values using two symbols: 0 and 1. Each digit in a binary number is called a bit.

Computers use binary because their internal circuitry (transistors and logic gates) can be in one of two states: on (1) or off (0).

The Binary Number System uses only two digits, 0 and 1, which are essential for computer operations. Each 'bit' represents one of these two states, allowing computers to process a variety of data. The hardware of computers, primarily made up of transistors and logic gates, works in binary because they can only be in one of two states, representing an electric signal being either on or off.

Think of a light switch. The light can either be on (1) or off (0). Similarly, binary works with these two states to represent complex data and instructions to the computer.

Binary numbers can be converted to decimal by expanding the binary number using powers of 2: Example: Convert the binary number 1101 to decimal. 1101_2=1×2^3+1×2^2+0×2^1+1×2^0=8+4+0+1=13_{10}

To convert a binary number to decimal, each bit is multiplied by 2 raised to the power of its position (counting from right to left, starting at 0). For instance, in the binary number 1101, the leftmost bit (1) represents 2^3, the next (1) represents 2^2, the next (0) represents 2^1, and the rightmost (1) represents 2^0. We then add these values together: 8 + 4 + 0 + 1 equals 13 in decimal.

You can think of this like a score in a game where different achievements count as different points. Achieving a harder level gives you more points, similar to how a higher position adds more value in binary conversion.

Decimal numbers can be converted to binary using successive division by 2, noting the remainders.

Example: Convert decimal number 13 to binary. 13÷2=6 remainder 1 6÷2=3 remainder 0 3÷2=1 remainder 1 1÷2=0 remainder 1 Reading the remainders from bottom to top gives 1101_2.

To convert a decimal number to binary, repeatedly divide the number by 2 and keep track of the remainders. Start with the decimal number and divide it; the first division gives you a quotient and a remainder. Continue dividing the quotient until you reach zero. The binary number is then formed by reading the remainders in reverse order—bottom to top.

Imagine you have 13 apples and you're giving them away in pairs. Every time you give away a pair, you note down that you still have some left until you run out. The remainders you noted down give you the complete picture (or binary representation) of how many pairs and singles you had.

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