Subtraction in base two


Subtraction in base two is similar to subtracting numbers in base ten. Our goal with this lesson is to first help you understand deeply subtraction in base ten.

Then, as you learn how to subtract in base two, the similarity with base ten will allow you to grasp the concept better

Base ten uses 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

How would you subtract 116 from 375 in base ten?

     3 7 5
   - 1 1 6
_____________


Since 5 is smaller than 6, you will borrow a ten from 7 tens.

7 tens is now 6 tens. Then, add the ten you borrowed to 5 to get 15

Rewrite the problem as shown below.

           6 15
     7 5
   - 1 1 6
_____________


Base 2 uses only 0 and 1

With base 2, you will borrow a 2,4,8,16, etc depending on the place value, not a 10 when needed. Why is that?

Some explanations:

In base 10, the number 739 can mean everything you see below:

7 × 102 + 3 × 10 + 9

7 groups of 100, 3 groups of 10, and 9

7 hundreds + 3 tens + 9

In base 2, the number 1012 could also mean everything you see below:

1 × 22 + 0 × 2 + 1

1 groups of 4, 0 groups of 2, and 1

1 four + 0 two + 1

Simply put, it is because you are in base 2, so any borrowing is done with 2, 4, 8, etc


Say you want to perform the following subtraction in base two

     1 1 0
   - 1 0 1
_____________


From the twos place, borrow 2 from 1 two. 1 two in now 0 two

Then, add that 1 two to the 0 in the ones place to make it 1 two or 2

Rewrite the problem and subtract

           0   2
     1 0
   - 1 0 1
_____________
     0 0 1


Now you are ready to do some more subtraction in base two

Example #2:No carry

To avoid confusion with base 10, we put a 2 next to each number

However, it is clear to you that the subtraction is being done in base two, there is no need to write down the 2

     1 1 12
   - 0 1 02
_____________________
      1 0 12


The subtraction above is easy since there was no carry



Example #2:Subtraction in base two with carry

     1 1 0 0 12
   - 1 0 1 1 12
_________________________________
     1 1 0 0 02


Explanation:

Subtract the numbers in the ones place: 1 - 0 = 0

Write 0 in the ones place

Now, you have a problem. You cannot subtract the numbers in the twos place, neither the numbers in the fours place

You need to borrow 1 eight from the eights place. 1 eight is now 0 eight in that place

However, 1 eight = 1 four + 2 twos

Add 1 four to 0 four in the fours place to make it 1 four

Add 2 twos to 0 two in the twos place to make it 2 twos

Rewrite the problem and finish the subtraction

               0    1 2
     1 1 0 0 12
   - 1 0 1 1 12
_________________________________
     0 0 0 1 02






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