# Binary and Decimal Conversion: A Complete Guide

Contents

A binary number is a number that consists of only two digits: 0 and 1. The binary number system is also known as the base-2 system because each digit represents a power of 2. For example, the binary number 1011 represents 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0, which is equal to 11 in decimal.

A decimal number is a number that consists of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The decimal number system is also known as the base-10 system because each digit represents a power of 10. For example, the decimal number 123 represents 1×10^2 + 2×10^1 + 3×10^0, which is equal to 123 in decimal.

Binary and decimal numbers are widely used in mathematics, computer science, engineering, and everyday life. Therefore, it is important to know how to convert between them. In this article, we will explain how to convert binary numbers to decimal numbers and vice versa, using simple methods and examples.

## What is a Binary Number?

A binary number is defined as a number that has only two possible values: 0 or 1. These values are also called bits, which stand for binary digits. A bit can represent two states: on or off, true or false, high or low, etc.

Binary numbers are used to store and process information in digital devices, such as computers, smartphones, calculators, etc. Binary numbers can also be used to encode data, such as text, images, audio, video, etc.

The advantage of using binary numbers is that they are easy to manipulate and operate with electronic circuits. For example, a binary addition can be performed by using simple logic gates, such as AND, OR, XOR, etc.

The disadvantage of using binary numbers is that they can be very long and cumbersome to read and write. For example, the decimal number 1000 requires only four digits in base-10, but it requires ten digits in base-2: 1111101000.

## What is a Decimal Number?

A decimal number is defined as a number that has ten possible values: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These values are also called digits. A digit can represent any value from zero to nine.

Decimal numbers are used to represent quantities and measurements in everyday life. For example, we use decimal numbers to count money, measure length, weight, time, temperature, etc.

The advantage of using decimal numbers is that they are familiar and easy to read and write. For example, the decimal number 1000 requires only four digits in base-10.

The disadvantage of using decimal numbers is that they are not compatible with digital devices. For example, a decimal addition cannot be performed by using simple logic gates.

## How to Convert Binary Numbers to Decimal Numbers?

To convert a binary number to a decimal number, we need to follow these steps:

• Identify the position and value of each bit in the binary number.
• Multiply each bit by its corresponding power of 2.
• Add up all the products to get the decimal equivalent.

Let’s see an example:

Convert the binary number (11010)2 to a decimal number.

Solution:

We can write the binary number as follows:

(11010)2 = (1×2^4) + (1×2^3) + (0×2^2) + (1×2^1) + (0×2^0)

We can multiply each bit by its corresponding power of 2:

(11010)2 = (16) + (8) + (0) + (2) + (0)

We can add up all the products to get the decimal equivalent:

(11010)2 = (26)10

Therefore,

(11010)2 = (26)10

## How to Convert Decimal Numbers to Binary Numbers?

To convert a decimal number to a binary number, we need to follow these steps:

• Divide the decimal number by 2 and get the quotient and the remainder.
• Write down the remainder as the rightmost bit of the binary number.
• Repeat the process with the quotient until it becomes zero.
• Write down the final quotient as the leftmost bit of the binary number.
• Read the binary number from left to right.

Let’s see an example:

Convert the decimal number (87)10 to a binary number.

Solution:

We can divide the decimal number by 2 and get the quotient and the remainder as follows:

We can write down the remainder as the bits of the binary number from right to left:

(87)10 = (1010111)2

Therefore,

(87)10 = (1010111)2

## How to Convert Decimal Fractions to Binary Fractions?

To convert a decimal fraction to a binary fraction, we need to follow these steps:

• Multiply the decimal fraction by 2 and get the integer part and the fractional part.
• Write down the integer part as the leftmost bit of the binary fraction.
• Repeat the process with the fractional part until it becomes zero or repeats.
• Write down the final integer part as the rightmost bit of the binary fraction.
• Read the binary fraction from left to right.

Let’s see an example:

Convert the decimal fraction (0.95)10 to a binary fraction.

Solution:

We can multiply the decimal fraction by 2 and get the integer part and the fractional part as follows:

We can write down the integer parts as the bits of the binary fraction from left to right:

(0.95)10 = (0.11111…)2

Therefore,

(0.95)10 = (0.11111…)2

## How do Convert Negative Decimal Numbers to Binary Numbers?

To convert a negative decimal number to a binary number, we need to follow these steps:

• Convert the absolute value of the negative decimal number to a positive binary number.
• Find the one’s complement of the positive binary number by flipping all its bits.
• Find the two’s complement of the positive binary number by adding one to its one’s complement.
• Write down a negative sign before the two’s complement.

Let’s see an example:

Convert the negative decimal number (-9)10 to a binary number.

Solution:

We can convert the absolute value of (-9)10 to a positive binary number:

|(-9)| = |9| = (1001)2

We can find the one’s complement of (1001)2 by flipping all its bits:

(1001)’ = (0110)

We can find the two’s complement of (1001)2 by adding one to its one’s complement:

(1001)” = (0110) + (0001) = (0111)

We can write down a negative sign before the two’s complement:

(-9)10 = -(0111)2

Therefore,

(-9)10 = -(0111)2

## Summary

In this article, we have learned how to convert binary numbers to decimal numbers and vice versa. We have also learned how to convert decimal fractions to binary fractions and vice versa. We have also learned how to convert negative decimal numbers to binary numbers using the two’s complement method.

We have seen that binary numbers are useful for digital devices and data encoding, while decimal numbers are useful for everyday life and measurements. We have also seen that conversion between binary and decimal numbers can be done by using simple arithmetic operations and logic rules.

We hope that this article has helped you understand the concepts of binary and decimal conversion better. If you have any questions or feedback, please let us know in the comments below. Thank you for reading!

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