Non-Vedic Maths: Half and five for the odds

In my series of Non-Vedic maths I will present mental arithmetic techniques that allow you to speed up your number-crunching. In contrast to Vedic maths, I will also explain why these techniques work. I believe that being able to add and multiply quickly does not make a good mathematician. But understanding how things work and being able to develop your own techniques is more important.

Enough already! I'm repeating myself!

Today's technique teaches how to multiply any number by 5, 50, 500 etc. The technique is based on the simple fact that

5 = 10/2

In this way multiplication by 5 is the same as multiplication by 10 and the division by 2. By using a technique for quickly dividing by 2 the method is accelerated further.

So let's start with an example. Say you want to multiply 3465 by 5. We start with multiplying by 10, i.e. adding a zero at the end

3465 × 5 = 34650 / 2

Dividing by two can be done in two ways. If you know your multiples of two up to 9×2=18 by heart then the traditional method is probably best. Just divide every digit from left to right including the carry over from the remainder. Note that the carry over is either 0 or 1, so the largest number that you will ever have to divide is 19. In our example we get

34650 / 2 = 17325

This is our result!

Some people are not as comfortable or quick with dividing numbers above 10 by 2. For these people the following alternative method might be faster. First divide every digit by 2 and forget about the remainder or carry over,

34650
12320

Next we shift the number to divide one place to the right (this was our original number) and replace every even digit with 0 and every odd digit with 5.

3465
5005

This second step effectively accounts for the carry overs that we missed in the first step. Now add the results of these two operations together

12320
 5005
-----
17325

Because all the digits of the first number result from division of numbers less than ten, none of the digits can be larger than 4. For this reason we never have to worry about carry-overs when adding the two numbers.

For a slightly more mathematical explanation to what we have done here, let's write 34650/2 in the following way

34650/2 = (24640 + 10010)/2
        = 24640/2 + 10010/2
        = 12320 + 5005
        = 17325

To summarise, in order to multiply a number by five

• append a zero at the right of the number
• divide every digit by two, ignoring any remainders
• add five to every digit to the right of an odd digit of the number that you got after the first step.

Non-Vedic Maths: One more here, one more there, do I care?

Multiplying two numbers whose last digits add up to powers of 10

This is my first post in the series of Non-Vedic maths, where I will be explaining speed calculation techniques. These techniques are NOT based on the ancient Indian Vedas but can be derived using elementary maths.

In this first instalment of the series I will be looking at how to multiply two numbers whose last digits add up to powers of 10 and whose leading digits are identical. To explain the method let us first look at two digit numbers. As an example, we multiply 63 with 67. For this method to work the first digit has to be the same for both numbers and the last digits have to add up to 10. Then we can write our to numbers as

10a + b and 10a + c

where b + c = 10 and a is the leading digit. In our example we have a = 6, b = 3 and c = 7. Multiplying the two numbers gives





In the third step we used that b+c=10. The product bc will always be less than 100 so the digits will not interfere with the first term, which is a multiple of 100. In our example we have

63 * 67 = 100 * 6 * 7 + 3 * 7 = 100 * 42 + 21 = 4221

The rule that can be extracted from this is:

Take the first digit and multiply with one more than itself. Multiply the last two digits. The final answer is made up of the two results by joining the results together.

We can represent this graphically in the following way.

The method clearly also works if a is not a single digit number, but it can have any number of digits. If we wanted to multiply 116 with 114 we will get

100 * 11 * 12 + 6 * 4 = 13200 + 24 = 13224.

The method can be generalised if the trailing two digits add up to 100 or the trailing three digits add up to 1000 and so on. The maths is similar to the previous case.



Again, bc will always be less than 10000 because each of them is less than 100. To see an example, lLet's multiply the numbers 436 and 464. Here the last two digits add up to 100 and the leading digit is the same in both numbers.



Of course one has to multiply two digit numbers, which can be a little more complicated. In another post I will present methods that can make this multiplication easier.

Non-Vedic Maths: An Introduction

In the past I have repeatedly been confronted with people who promote Vedic Mathematics. Vedic Maths is a system of mathematical procedures that is supposed to help you master mathematics. The people promoting this system claim that it is based on the old Indian Vedic Sutras which date back over 4000 years. A little study of the subject reveals a slightly different story. Vedic mathematics was first presented by an Indian mathematician called Bharati Krishna Tirthaji Maharaja, in the early part 1900s.

He claimed to have found uncovered the system by intensely studying the old Vedas. However, most scholars, that are not directly involved in promoting the system, agree that the Vedas do not actually contain any of the "Vedic mathematics" sutras. It is, therefore, much more likely that a Tirthaji invented the methods himself, or even copied them from other sources. Another argument against the origin from the Vedas is that the techniques described in Vedic maths heavily rely on the decimal number system. But this system was not invented until much later, after the Vedas were written down.

The Vedic maths system claims to provide calculation strategies which are creative and useful. But they don't really promote a deeper understanding of mathematics. All that these methods really do, is to provide the student with some shortcuts for performing standard calculations in their head. It is interesting to note that the mathematics behind the system is not very sophisticated and was certainly known at the time that a Tirthaji developed these methods.

So, quite clearly, Vedic maths does not date back to Vedic times and cannot be extracted from the Vedas. It is a modern invention that uses the claim of dating back to ancient times in order to promote itself. But all this doesn't answer the crucial question, is the Vedic maths system useful? As I said before, the methods provided do not really promote analytical thinking and, in todays world of computers and pocket calculators, they are not needed in everyday life, and they will not help the student in their future job. To make matters worse, the Vedic mathematics system does not explain why or how the techniques work. So the student is left with learning the methods by heart and repeat blindly. If you believe that education should involve more than pure repetition of facts, and you don't want your children to grow up being parrots, then avoid the Vedic maths system. For two more passionate articles against the system, you can read The Fraud of Vedic Maths or Stop this Fraud on our Children!

Arithmetic shortcuts do have their use however, and that is to impress your friends. If you can find the result of a complex product or sum before your friends can get out their iPhones and start the calculator app, then they will be truly impressed. And if you can still pull this feat after a few pints in the pub, you will be considered the maths genius for the rest of your life. For this reason alone, I will start a series of posts under the heading "Non-Vedic Maths". This series will present mathematical  parlour tricks and shortcuts for everyday calculations to impress your friends. I will also go the extra mile and explain why the techniques work. But be warned: learning these methods will not help you in any other way, and you will most certainly be branded a geek by everyone around you.

Read on for the first technique.