How to Subtract Two Binary Numbers

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It’s a little different, but with the steps below, subtracting two binary numbers will be just as simple, or even simpler, than subtracting two ordinary decimals.

Table of Contents

Use the borrowing method

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Set the calculation as usual. Big numbers are placed above small numbers. As with decimals (base 10), if the smaller number has fewer characters or digits, the numbers will be written in the right margin. ^{[1] X Research Source}

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Solve some basic math problems. For some problems with subtracting two binary numbers, the solution is no different from what is done in the base ten system. We just do the math and find each digit of the difference, starting from right to left. Here are some simple examples:

1 – 0 = 1
11 – 10 = 1
1011 – 10 = 1001

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Experiment with more complex problems. You only need to know a special “rule” to be able to solve any problem except two binary numbers. This rule says we need to “borrow” from the left digit to compute the column “0 – 1”. ^{[2] X Source of Research} In the remainder of this section, we will set up a few example problems and solve them using the borrowed method. Example 1:

110 – 101 = ?

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“Borrow” from the second digit. Starting from the right column (position one), we need to find “0 – 1”. To subtract, we must “borrow” from the left digit (position two). Here are two steps:

First, cross out 1 and replace it with 0, we get: 1 ⁰1 0 – 101 = ?
We’ve just taken away 10 from the first number so we can add this “borrowed” value to position one: 1 ⁰1 ¹⁰0 – 101 = ?

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Solve the rightmost column. Now any column can be subtracted as usual. With the rightmost column (position one) in this example: ^{[3] X Research Source}

1 ⁰1 ¹⁰0 – 101 = ?
The rightmost column will now be: ¹⁰ – 1 = 1. If you don’t understand how we got to this answer, you can convert it to decimal:
10 ₂ = (1 x 2) + (0 x 1) = 2 ₁₀ . (The _index number indicates what the base of the number is.)
1 ₂ = (1×1) = 1 ₁₀ .
Therefore, the decimal form of this problem will be 2 – 1 = ?, answer: 1.

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Complete the quiz. We can finish the rest easily. Subtract each column, in order from right to left:

1 ⁰1 ¹⁰0 – 101 = __1 = _01 = 001 = 1 .

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Try harder math problems. Borrowing is used a lot in binary multiplication, and sometimes, you will have to borrow many times just to subtract a column. Example 2: 11000 – 111 . Since we can’t “borrow” from 0, we have to keep borrowing to the left until we reach a value we can borrow: ^{[4] X Research Source}

1 ⁰ 1 ¹⁰ 0 00 – 111 =
1 ⁰ 1 ¹10 0 ¹⁰0 0 – 111 = (don’t forget 10 – 1 = 1)
1 ⁰ 1 ¹10 0 ^¹100 ¹⁰0 – 111 =
Briefly write: 1011 ¹⁰0 – 111 =
Solve each column in turn : _ _ _ _ 1 = _ _ _ 0 1 = _ _ 0 0 1 = _ 0 0 0 1 = 1 0 0 0 1

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Check answers. There are three ways to check your answers. ^{[5] X Research Source} You can do a quick check by going online to find a binary calculator and then enter the calculation. Or you can use the two methods below, they will help you get familiar with and proficient with binary numbers and therefore, will be very useful when you can’t rely on a calculator for the exam.

Add them together to check your answers. When adding the difference for the smaller number, the result must be the large number. With example 2 (11000 – 111 = 10001), we have 10001 + 111 = 11000, which is the original larger number.
Or, you can convert each binary number to decimal and check if the calculation is correct. With the above example (11000 – 111 = 10001), we can convert each number to decimal and have the equation 24 – 7 = 17. This equation is correct, and therefore, our answer is correct.

Using the complement method

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Set the calculation as usual. Used by computers to subtract binary numbers, this method uses a more efficient program. For those familiar with the usual subtraction of two decimals, this approach can be a bit more difficult to use. However, if you are a programmer, you should still understand this method of offset because it can be very useful. ^{[6] X Research Source}

Example problem: 101 – 11 = ?

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Add 0 in front so that both numbers have the same number of digits. For example, convert 101-11 to 101-011 so that both binary numbers have three digits.

101 – 011 = ?

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Convert the digits in the following number. Convert every 0 to 1 and every 1 to 0. In the example problem, the second number becomes: ~~011~~ → 100 .

Actually, we’re “taking the complement” or subtracting each digit from one by one. This shortcut “conversion” works for binary numbers because the conversion only gives two possible outcomes: 1 – 0 = 1 and 1 – 1 = 0 .

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Add one to the newly formed number. Having “reversed” the second number, we will now add one to the result. In the example problem, we have 100 + 1 = 101 .

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Solve new problems like binary addition. Instead of subtracting, this time, we will use binary addition to add the new number to the original number:

101 + 101 = 1010
If you don’t understand, you can review how to add two binary numbers.

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Remove the first digit. The answer found in this method always has one digit left over. In the example under consideration, the two numbers initially have only three digits (101 + 101), but the answer has four digits (1010). We just cross out the first digit, and the result is the answer of the original subtraction : ^{[7] X Research source}

1 010 = 10
So 101 – 011 = 10
If the answer has no extra digits, you must have subtracted the large number from the smaller number. Refer to the tips section for how to solve such problems and try again.

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Try this method in base 10. This method is also known as the “two’s complement” method because the “inverting” step results in “one’s complement” and an additional step of 1. ^{[8] X Source} If If you want to understand this method more intuitively, you can try it in base ten:

56 – 17
Since we use the base ten system, we’ll get the “nine’s complement” from the following number (17) by subtracting nine from each of the digits. 99 – 17 = 82 .
Convert to addition: 56 + 82 . Comparing with the original problem (56 – 17), we can see that we have added 99.
56+82= 138 . But since we added 99 to the original math, now we need to subtract 99 from the answer. Here, we shorten again as in the method with binary numbers above: add 1 to the sum and then cross out the digit to the left (representing 100):
138 + 1 = 139 → 1 39 → 39 . This is the final answer to the original problem, 56-17.

Advice

To subtract small numbers from large numbers, swap their positions, perform the subtraction, and add a minus sign in front of the answer. For example, to solve the binary problem 11 – 100, we find the answer of 100 – 11 and then add a minus sign in front of the answer (this rule applies to all base systems, not just binary).
Mathematically, the complement method applies the equation a – b = a + (2 ⁿ – b) – 2 ⁿ Where n is the number of digits of b, 2 ⁿ – b is equal to 1’s complement.

Steps

Use the borrowing method

Using the complement method

Advice

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