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Foundations-Multiplying by Half of Ten

The Multiplication Workhorse




 Let's begin again with a problem. Can you easily multiply 484 × 5 in your head? If so, then how about 3261 × 5? By the end of this post, you will be able to easily multiply any number by five provided you have mastered doubling and halving discussed in my previous posts.

 Multiplication by five is the workhorse of martian mathematics. At least it is for the intermediate methods of multiplying times 3, 4, 5, 6, and 7. More on that in a later post. I think that everyone would agree that aside from multiplying by one, nothing is easier than multiplying by ten—just add a zero onto the end of the number. For example, 37 × 10 = 370, 271 × 10 = 2710, and hell × 10 = hell0. Well, maybe not that last one, but you get the idea: just tack a zero onto the back of the number when multiplying by ten. 

 Multiplying by five is almost as easy as multiplying by ten because five is just half of ten. 

Using the commutative property of multiplication (a × b = b × a)—commute means to travel—we can move the division by two over to the a term.

Using the notation from the halving post we can write this as:

Then we can think of 4 × 5 as half of 40 or 20 and 86 × 5 as half of 860 or 430. It helps to imagine a zero after the number you are multiplying by five and simply halve that. Let's try a few examples.


The red dots are placed after each odd digit to remind us to wear our conjugation goggles going forward, and a* means the five's conjugate of a. If any of this is unfamiliar, then read the posts in the links. 

 Let's revisit the problems at the beginning of the post. Can you do them now? Half of 4840 is 2420 and half of 32610 is 16305. Sometimes, it is more economical to handle more than one digit at a time. You probably recognized that half of 32 is 16 and that half of 610 is 305. Look for groupings like this to save even more time. Here are some more problems to practice on.


In my next post, we will explore the inverse of multiplication by five—dividing by five. 
© 2019 David W. Ward, All Rights Reserved.
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