Binary to Decimal

Understanding Binary and Decimal Number Systems

When it comes to understanding computer systems, one of the fundamental concepts is the binary number system. Computers at their core work with binary numbers - ones and zeros. However, for humans, it is much easier to work with decimal numbers - numbers from 0 to 9. In this article, we will explore the relationship between binary and decimal numbers and learn how to convert between the two.

What is Binary?

Binary is a number system that uses only two digits - 0 and 1. In contrast to the decimal system, where each digit represents a multiple of ten, in binary, each digit represents a power of two. The rightmost digit represents 2^0 (which is equivalent to 1), the next digit represents 2^1 (which is equivalent to 2), the next digit represents 2^2 (which is equivalent to 4), and so on. Binary numbers are read from right to left, just like decimal numbers.

Counting in Binary

Counting in binary is quite different from counting in decimal. In the decimal system, when we reach 9, we carry over to the next place value, which is 10. However, in binary, when we reach 1, we carry over to the next place value, which is 10 (two in decimal). The counting sequence in binary looks like this: 0, 1, 10, 11, 100, 101, 110, 111, and so on.

What is Decimal?

Decimal is a number system that uses ten digits - 0 to 9. Each digit in a decimal number represents a power of ten. The rightmost digit represents 10^0 (which is equivalent to 1), the next digit represents 10^1 (which is equivalent to 10), the next digit represents 10^2 (which is equivalent to 100), and so on. Decimal numbers are read from right to left, with each digit being multiplied by its corresponding power of ten.

Conversion from Binary to Decimal

Converting a binary number to decimal is relatively straightforward. To convert a binary number to decimal, we need to multiply each digit of the binary number by its corresponding power of two and then sum the results. Let's take a look at an example to illustrate this process:

Example: Convert the binary number 1101 to decimal.

To convert the binary number 1101 to decimal, we multiply each digit by its corresponding power of two:

- The rightmost digit, which is 1, represents 2^0 (which is equal to 1).

- The next digit, which is also 1, represents 2^1 (which is equal to 2).

- The third digit from the right, which is 0, represents 2^2 (which is equal to 4).

- The leftmost digit, which is 1, represents 2^3 (which is equal to 8).

Now, we sum the results: 1 + 2 + 0 + 8 = 13.

Therefore, the binary number 1101 is equal to the decimal number 13.

Conversion from Decimal to Binary

Converting a decimal number to binary is a bit more complex than converting from binary to decimal. To do this, we need to repeatedly divide the decimal number by two and keep track of the remainders. The binary representation of the decimal number is obtained by writing down the remainders in reverse order. Let's look at an example to understand the process better:

Example: Convert the decimal number 27 to binary.

To convert the decimal number 27 to binary, we divide the number by two and keep track of the remainders until the quotient becomes zero:

- 27 divided by 2 is 13, with a remainder of 1 (the rightmost digit).

- 13 divided by 2 is 6, with a remainder of 1.

- 6 divided by 2 is 3, with a remainder of 0.

- 3 divided by 2 is 1, with a remainder of 1.

- 1 divided by 2 is 0, with a remainder of 1 (the leftmost digit).

Now, we write down the remainders in reverse order: 11011.

Therefore, the decimal number 27 is equal to the binary number 11011.

Binary and Decimal in Computing

Binary is the most fundamental number system used in computing. Computers use binary numbers internally to represent all forms of data, including text, images, audio, and video. Every piece of information stored or processed by a computer is ultimately represented in binary. However, for human interaction, computers often present data in decimal form. This is because most people are more comfortable working with decimal numbers and find them easier to understand.

Summary

Binary and decimal are two number systems used by computers and humans, respectively. Binary uses two digits (0 and 1) and represents each digit as a power of two. Decimal uses ten digits (0 to 9) and represents each digit as a power of ten. Converting between binary and decimal is relatively straightforward, requiring multiplication and addition for conversion in both directions. Binary is the fundamental number system used in computing, while decimal is the most commonly used number system for human interaction. Understanding the relationship between binary and decimal is essential for anyone working in the field of computer science or computing technology.