1. Numbers

What are Numbers?

Numbers are fundamental to computing and are used to represent quantities, values, or positions. In computer applications, numbers play a critical role in performing mathematical operations, storing data, and driving logical decisions in programs.

Numbers are the building blocks of computing. They represent various values and are essential for a computer to perform tasks. When we input data into a computer, whether it's counting items, performing calculations, or determining the position of objects in a program, numbers are involved in these processes. Without numbers, computers would not be able to execute programs, process information, or make decisions.

Think of numbers in computing like ingredients in a recipe. Just as a recipe needs ingredients to create a dish, computers need numbers to perform functions and operations to provide outputs.

Number Systems in Computing

In computing, numbers are represented in different number systems. The most common ones are:

• Decimal Number System (Base 10): The standard number system used in everyday life, consisting of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Binary Number System (Base 2): The fundamental number system in computing, where only two digits (0 and 1) are used.
• Octal Number System (Base 8): Uses digits 0 to 7 and is often used in computing to simplify binary representation.
• Hexadecimal Number System (Base 16): Uses 16 digits (0-9 and A-F) and is widely used in programming and memory addressing.

In computing, different number systems are used based on the requirements and efficiency of data processing. The decimal system is the one most people use daily, but computers primarily work with binary, which consists only of zeros and ones. The octal system condenses binary numbers for easier reading, while hexadecimal allows concise representation of binary numbers using a base-16 system, making it convenient for programmers working with large amounts of data.

You can think of different number systems like languages. In your daily life, you speak a language (decimal), but a computer speaks its unique language (binary). Just as various languages are adapted for different purposes, different number systems serve specific needs in the computing world.

What is Binary?

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 system only uses the digits 0 and 1, which corresponds to the two states of a computer's hardware. Each bit represents a simple state of being either off or on. This simplicity allows computers to perform complex calculations and operations efficiently, as they can easily transcribe and manipulate data in these two states.

You can imagine a light switch in your house. The 'off' position represents 0, and the 'on' position represents 1. Just as you can combine multiple switches to control lighting in a room, a computer combines many bits to process and store information.

Converting Binary to Decimal

Binary numbers can be converted to decimal by expanding the binary number using powers of 2:

Example: Convert the binary number 1101 to decimal.

$1101_2 = 1\times 2^3 + 1\times 2^2 + 0\times 2^1 + 1\times 2^0 = 8+4+0+1=13_{10}$

To convert from binary to decimal, you multiply each bit by its corresponding power of 2 and then sum the results. Starting from the right, the first bit is multiplied by 2 raised to the power of 0, the second bit by 2 raised to the power of 1, and so forth. By calculating this for each bit and adding the results, you can easily find the decimal equivalent.

Imagine you’re counting points in a game where each position has a different score depending on how far you are from the starting point. Here, each bit in binary corresponds to an increasing score based on its position, similar to how your score increases more the further you score from the origin.

Converting Decimal to Binary

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

Example: Convert decimal number 13 to binary.

$13\div2=6 \text{ remainder } 1\newline 6\div2=3 \text{ remainder } 0\newline 3\div2=1 \text{ remainder } 1\newline 1\div2=0 \text{ remainder } 1$

Reading the remainders from bottom to top gives 1101_2.

To convert a decimal number to binary, divide the number by 2 repeatedly, keeping track of the remainders at each step. The result is read from the last remainder obtained to the first, yielding the binary representation. This method leverages the binary system's base-2, ensuring that each number is represented accurately.

Consider baking a cake where you trim layers off to get to the final shape. Each division represents trimming, and you note what you have removed (the remainders). When you put together the layers from bottom to top after removing them, you get the final cake shape — just like reading remainders gives you the final binary representation.

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