Duplication algorithm

The best way to understand the duplication algorithm is with examples along with some explanations.  

Technique:

To multiply 8 and 12, keep multiplying 12 by power of 2 until the last power of 2 is 8.

Power of 2 are 2, 4, 8, 16 and so forth. 

Here is how you can multiply 8 and 12.

2 × 12 = 24

4 × 12 = 48

8 × 12 = 96

All you do here is a succession of doubling operations. Every time you double 2, you also double the number on the right of the equal sign. Now that you got the technique, we can do some more interesting multiplication problems.

More examples showing how the duplication algorithm works


Example #1:

14 × 12

If doubling 8 gave 14, we could just do that and then just double 96 to get the answer. However, doubling 8 gives 16, not 14. Therefore, we have to find a way to play with the numbers to get what we need.

Notice that 14 = 8 + 4 + 2, so 14 × 12 = ( 8 + 4 + 2 ) × 12 = 8 × 12 + 4 × 12 + 2 × 12

We already have answers for 8 × 12, 4 × 12, and 2 × 12

All we need to do is to add 24, 48, and 96

24 + 48 + 96 = 168

14 × 12 = 196

Example #2:

26 × 20

Solution:

2 × 20 = 40

4 × 20 = 80

8 × 20 = 160

16 × 20 = 320

32 × 20 = 640

Again, we cannot really use 32 × 20 = 640

We need to rewrite 26 × 20

26 × 20 = (16 + 8 + 2) × 20 = 16 × 20 + 8 × 20 + 2 × 20

We already have answers for 16 × 20 + 8 × 20 + 2 × 20

320 + 160 + 40 = 320 + 200 = 520


Example #3:

48 × 25

Solution:

2 × 25 = 50

4 × 25 = 100

8 × 25 = 200

16 × 25 = 400

32 × 25 = 800

64 × 25 = 1600

For the last time, we cannot really use 64 × 25 = 1600

We need to rewrite 48 × 25

48 × 25 = (32 + 16) × 25 = 32 × 25 + 16 × 25

We already have answers for 32 × 25 + 16 × 25

800 + 400 = 1200

What is the point of using the duplication algorithm? May be doubling numbers is a friendly way of doing multiplication. Sometimes, it may simplify the problem yet other times it may complicate the problem.

It is up to you what you use, but it is a good thing to know how to use the duplication algorithm.It is one more tool at your disposal when performing multiplication!

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