Multiplication in base two

Multiplication in base two could be a daunting task. However, my goal is to really provide you with crystal clear explanations.

The algorithm to multiply numbers in base 2 is quite similar to that in base 10.  

If you do not understand the example below, do not sweat!

Keep reading and check out example #1 to see how we broke it down into smaller manageable steps.

Plan to spend some time reading this lesson about multiplication in base two carefully and without distraction.

Difference between base ten and base two

Take a close look at the difference between base 2 place value and base 10 place value shown below. The difference is shown with two numbers: 97462 and 10011₂

For 97462, 7 is in the thousands place and it means there are 5 thousands

For 10011₂, starting from right to left, the last 1 is in the sixteens place and it means there is 1 sixteen.

Ten thousands	Thousands	Hundreds	tens	ones
9	7	4	6	2

sixteens	eights	fours	twos	ones
1	0	0	1	1

Ready to do multiplication in base two?

Example #1:

         1 1 0 1₂
      ×       1 1₂
   ________________


First multiplication:

         1 1 0 1₂
      ×          1₂
   ______________
         1 1 0  1₂


Explanations:

Starting from right to left,

1 × 1 = 1. Write down 1 in the ones place. (shown in red)

1 × 0 two = 0 two. Write down 0 in the twos place. (shown in green)

1 × 1 four = 1 four. Write down 1 four in the fours place. (shown in blue)

1 × 1 eight = 1 eight. Write down 1 eight in the eights place. (shown in brown)


Second multiplication:

         1 1 0 1₂
      ×       1₂
  ______________
    1  1 0  1 0₂


Explanations:

Notice that there is no digit in the ones place next to the 1. It is because we already performed this multiplication above.Therefore, make sure you put a 0 in the ones place as shown in red.

Starting from right to left,

1 two × 1 = 1 two. Write down 1 in the twos place. (shown in green)

1 two × 0 two = 0 four. Write down 0 in the fours place. (shown in blue)

1 two × 1 four = 1 eight. Write down 1 eight in the eights place. (shown in brown)

1 two × 1 eight = 1 sixteen. Write down 1 sixteen in the sixteens place. (shown in black)

Add the two results. Before learning how to do multiplication in base two, you should review the lesson about adding in base two.


             1  1  0  1₂
        1  1  0  1  0₂
__________________
    1  0  0  1  1  1₂


Example #2:

         1 0 1 0₂
      ×       1 0₂
   ________________


First multiplication:

         1 0 1 0₂
      ×          0₂
   ________________
        0  0 0 0₂


Explanations:

Starting from right to left,

0 × 0 = 0. Write down 0 in the ones place. (shown in red)

0 × 1 two = 0 two. Write down 0 in the twos place. (shown in green)

0 × 0 four = 0 four. Write down 0 four in the fours place.(shown in blue)

0 × 1 eight = 0 eight. Write down 0 eight in the eights place. (shown in brown)


Second multiplication:

         1 0 1 0₂
      ×       1₂
   ________________
   1  0  1  0  0₂


Explanations:

Notice that there is no digit in the ones place next to the 1. It is because we already performed this multiplication above. Therefore, make sure you put a 0 in the ones place as shown in red.

Starting from right to left,

1 two × 0 = 0 two. Write down 0 in the twos place.(shown in green)

1 two × 1 two = 1 four. Write down 1 in the fours place. (shown in blue)

1 two × 0 four = 0 eight. Write down 0 eight in the eights place. (shown in brown)

1 two × 1 eight = 1 sixteen. Write down 1 sixteen in the sixteens place. (shown in black)

Add the two results.


              0   0  0  0₂
         1  0   1  0  0₂
_______________________________
         1  0   1  0  0₂

Again, it is quite challenging to do multiplication in base two. I recommend you take your time when reading this lesson. Read the lesson multiple times if needed. You will get it!