How to Count in Binary: 11 Steps (with Pictures)

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

Want to boost your brainpower so you can wow your nerdy friends? Learn how the binary system works, which is the basis of the operation of any modern electronic device (computer, video game console, smartphone, tablet, etc.). At first, accustomed to the decimal system, counting in binary may seem strange to you, but with a little practice and a few simple rules to follow you will learn in no time.

Reference Table

Decimal System	0	1	2	3	4	5	6	7	8	9	10
Binary System	0	1	10	11	100	101	110	111	1000	1001	1010

Steps

Part 1 of 2: Discovering the Binary System

Step 1. Learn the basics of the binary numbering system

The set of numbers that is normally used by all humans is called the decimal system or, more technically, the "base ten" system. This name derives from the fact that the decimal system is made up of 10 symbols which are used to represent all numbers and are between 0 and 9. The binary or "base two" system has only two symbols: 0 and 1.

Step 2. To add a unit in binary just change the least significant digit from 0 to 1

This rule only applies if the last digit to the right of the number under consideration is a 0. You can use this step to count the first two numbers of the binary system, exactly as you would expect to do:

• 0 = zero.
• 1 = one.

• In the case of larger numbers you will simply have to ignore the most significant digits and always refer to the least significant one. For example 101 0 + 1 = 101

  Step 1..

Step 3. If all digits of the number under consideration are equal to 1, you will need to add another one

Normally in this case we would have to use another symbol to count to two, but the binary system only predicts 0 and 1, so how do you proceed? Simple, add a new digit (with value 1) to the extreme left of the number and set all the others to 0.

• 0 = zero.
• 1 = one.
• 10 = two.
• This is the same rule that is also used by the decimal system when the symbols to represent numbers are exhausted (9 + 1 = 10). The only difference is that in the binary system this scenario is much more frequent, since there are only two symbols to use.

Step 4. Use the rules described so far to count to five

At this point you should be able to count from zero to five in binary in total autonomy, so give it a try and then check the correctness of your work using this scheme:

• 0 = zero.
• 1 = one.
• 10 = two.
• 11 = three.
• 100 = four.
• 101 = five.

Step 5. Count to six

Now we need to calculate the result given by the sum of five plus one, which in binary becomes 101 + 1. The key to doing this is to ignore the most significant digit, which is the one on the far left. Simply add 1 to the least significant digit and get 10 as a result (remember this is like writing 2 in binary). Now enter the most significant digit in its rightful place to get:

110 = six

Step 6. Count to ten

At this point you no longer need to learn other rules: you already have everything you need, so try to count to ten on your own. At the end check the correctness of your work using this scheme:

• 110 = six.
• 111 = seven.
• 1000 = eight.
• 1001 = nine.
• 1010 = ten.

Step 7. Note when you need to add a new digit to the previous number

Have you noticed that, unlike the decimal system, ten (1010) does not represent a "special" number? In binary it is the number eight (1000) that is much more important because it is the result of 2 x 2 x 2. Continue to calculate the powers of two to find the other relevant numbers in the binary system, such as sixteen (10000) and the thirty-two (100,000).

Step 8. Practice using larger numbers

Now you know all the rules to use for counting in binary. If you are unsure which is the next binary number, always refer to the value assumed by the least significant digit (the one on the far right). Here are some examples that should shed some light:

• Twelve plus one = 1100 + 1 = 1101 (0 + 1 = 1 and all other digits remain unchanged).
• Fifteen plus one = 1111 + 1 = 10000 that is sixteen (in this case we have exhausted the symbols of the binary system, so we add a new digit to the left and "reset" all the others).
• Forty-five plus one = 101101 + 1 = 101110 that is forty-six (as you know 01 + 1 = 10 while all the other digits remain unchanged).

Part 2 of 2: Converting a Binary Number to Decimal

Step 1. Take note of the position occupied by the single digits that make up the binary number to be converted

By learning to count in decimal, you have also learned the meaning assumed by each digit based on the position it occupies: units, tens, hundreds, thousands and so on. Since the binary system has only two symbols, the position taken by each single digit represents a power of two, whose index increases as it moves to the left:

• Step 1. is in first position (2⁰=1).
• Step 1.0 is in second position (2¹=2).
• Step 1.00 is in fourth position (2²=4).
• Step 1.000 is in eighth position (2³=8).

Step 2. Now multiply each digit of the number to be converted by the value corresponding to its position

Start with the least significant digit, the one on the far right, and multiply its value (0 or 1) by one. Now, on a new line, multiply the second digit value by two. Repeat this operation for all the digits that make up the binary number to be converted, continuing to multiply the relative value by the respective occupied position (i.e. by the corresponding power of two). Here is an example that will help you understand the mechanism:

• What is the decimal equivalent of the binary number 10011?
• The rightmost digit is a 1. This is the first position, so we will multiply its value by 1 to get: 1 x 1 = 1.
• The next digit is still 1. In this case it is in second position, so we will multiply it by two to get: 1 x 2 = 2.
• The next digit is 0 and is in the fourth position, so we will get: 0 x 4 = 0.
• The next digit is still 0 and is in the eighth position, so we will have: 0 x 8 = 0.
• The most significant digit is equal to 1 and is in the sixteenth position, so we will get: 1 x 16 = 16.

Step 3. Now add up all the partial results you have obtained

Now that we have converted every single binary digit into the corresponding decimal, to calculate the final value we simply add the single products together. Following the previous example we will get:

• 1 + 2 + 16 = 19.
• The binary number 10011 corresponds to the decimal number 19.