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Binary Number System: Application & Advantages

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  • 0:03 What Is the Binary System?
  • 0:56 Applications
  • 2:00 Advantages
  • 2:57 Lesson Summary
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Lesson Transcript
Instructor
David Karsner

David holds a Master of Arts in Education

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

All computers use a binary number system, that relies upon just two symbols, typically 0 and 1. In this lesson, you'll learn about the advantages of using the binary number system, as well as some of its practical applications.

What Is the Binary System?

There is a bumper sticker out there that reads:

'There are 10 kinds of people, those who understand binary and those who don't.'

If you don't already understand why this is funny, you will by the end of this lesson.

The most commonly used number system, called base-ten, uses ten digits: 0-9. By comparison, the binary number system, or base-two, is a counting technique that uses two digits: 0 and 1. Here, the prefix 'bi' means 'two.'

In this system, each place value is a power of two, where the first place to the left of the decimal point is 2^0, the second place is 2^1 and so on. Each number is called a bit and is pronounced separately. For example, when referring to this binary number:

1011

We'd say 'one zero one one.'

Applications

The most common application for the binary number system can be found in computer technology. All computer language and programming is based on the 2-digit number system used in digital encoding. Digital encoding is the process of taking data and representing it with discreet bits of information. These discreet bits consist of the 0s and 1s of the binary system.

For example, the images you see on your computer screen have been encoded with a binary line for each pixel. If a screen is using a 16-bit code, then each pixel has been told what color to display based on which bits are 0s and which bits are 1s. As a result, 2^16 represents 65,536 different colors!

We also find the binary number system in a branch of mathematics known as Boolean algebra. This field of mathematics is concerned with logic and truth values. Here, statements that are either true or false are then assigned a 0 or 1.

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Additional Activities

Converting Numbers from Binary to Decimal

If you are given a number written in binary, you can translate it to a decimal by multiplying each binary bit by its power of two, and adding all the results. For example, the binary number 11101 converted to decimal is (1*(2^4)) + (1*(2^3)) + (1*(2^2)) + (0*(2^1)) + (1*(2^0)) = 16 + 8 + 4 + 0 + 1 = 29. In the following examples, convert the binary number to its equivalent decimal number.

Examples

  • 1001
  • 10101
  • 11000

Solutions

  • For 1001 we have (1 * (2^3)) + (0 * (2^2)) + (0*(2^1)) + (1*(2^0)) = 8 + 1 = 9
  • For 10101 we have (1 * (2^4)) + (0 * (2^3)) + (1 * (2^2)) + (0 * (2^1)) + (1 * (2^0)) = 16 + 4 + 1 = 21
  • For 11000 we have (1 * (2^4)) + (1 * (2^3)) + (0 * (2^2)) + (0 * (2^1)) + (0 * (2^0)) = 16 + 8 = 24

Converting Numbers from Decimal to Binary

We can also convert numbers written in decimal to binary numbers. To do this, first find the largest power of 2 that is less than or equal to the decimal number. Find the difference between the original number and this power of 2. Then find the largest power of 2 that is less than or equal to this new number. Repeat this process until you have no powers of 2 left. The decimal number is the sum of all the powers of 2 you found - to write the number in binary, put a 1 in for the bit representing each power of 2 you found, and a 0 in for all other bits. For example, for the decimal number 29, the highest power of 2 that is less than or equal to 29 is 2^4 = 16. The difference between 29 and 16 is 13. The highest power of 2 less than or equal to 13 is 2^3 = 8. The difference between 13 and 8 is 5. The highest power of 2 that is less than or equal to 5 is 2^2 = 4. The difference between 5 and 4 is 1. The highest power of 2 less than or equal to 1 is 2^0 = 1. This last difference is 0, so we are done. This means our decimal number of 29 can be written as (1*(2^4)) + (1*(2^3)) + (1*(2^2)) + (0*(2^1)) + (1*(2^0)) which makes the binary version of 29 the number 11101. In the following examples, convert the decimal number to binary.

Examples

  • 19
  • 33

Solutions

  • For 19, the highest power of 2 is 2^4 = 16. The difference is 3. The highest power of 2 for 3 is 2^1 = 2. The difference is 1. The last power of 2 needed is 2^0 = 1, and the last difference is 0. This means that the decimal number 19 can be written as (1*(2^4)) + (0*(2^3)) + (0*(2^2)) + (1*(2^1)) + (1*(2^0)) and so the binary version of 19 is 10011.
  • For 33, the highest power of 2 is 2^5 = 32. The difference is 1. The highest power of 2 for 1 is 2^0 = 1. The difference is 0. This means that the decimal number 33 can be written as (1*(2^5)) + (0*(2^4)) + (0*(2^3)) + (0*(2^2)) + (0*(2^1)) + (1*(2^0)) thus the binary version of 33 is 100001.

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