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# Is it true that 19 x 21 = 20 x 20?

February 3, 2017

I recently came across a neat product called the QAMA Calculator. It is a calculator that encourages thought. Ask it to calculate something and QAMA will ask you back: “First give me an estimate”. Once you supply an approximate answer, QAMA calculates the precise answer for you. But only if your approximation was sensible. This calculator forces you to think.

Why use the QAMA calculator? Why think about calculations when calculators do the job just fine?

QAMA’s philosophy is that thinking about calculation estimates helps you improve your thinking ability, not just in mathematics but also in other domains. Using QAMA helps you understand the numbers at hand and get brain exercise on the go. As you or your loved ones use QAMA, confidence, ability and understanding improves.

In my case, a few hours with QAMA got me thinking about the following:

Is it true that 19 x 21 = 20 x 20?

After all, 19 is 1 less than 20 and 21 is 1 more. So maybe it all cancels out as happens for

19 + 21 = 20 + 20.

Me: 19x21.

QAMA: Give me an estimate.

Me (thinking): Well, 19 and 21 are both very close to 20. So I’ll compute 20 x 20 in my head as an estimate for QAMA.  Calculating 20 x 20 is easier

(2 x 10) x (2 x10) = (2 x 2) x (10 x 10) = 400.

Me (punching in my estimate to QAMA): 400.

QAMA (giving precise result): 399.

Me (feeling): Kind of nice! Even though 19 x 21 is not 20 x 20, the difference is only1.

Kind of interesting that the difference between my estimate and the exact answer is 1? Let’s think some more…. Or you know what, let’s just experiment…

Me: 29 x 31.

QAMA: Give me an estimate.

Me: 30 x 30 = (3 x 3) x 100 = 900.

QAMA: 899.

Me: 39 x 41.

QAMA: Give me an estimate.

Me: 40 x 40 = 1,600.

QAMA: 1,599.

I’m starting to think there is some pattern here!  OK, one more try:

Me: 15 x 17.

QAMA: Give me an estimate.

Me: I know that 16 x 16=256 because these values are often used computers. So my estimate is 256.

QAMA: 255.

The pattern that emerges is quite interesting. Here is one way to describe it:

If you have a square plot of land, Z meters by Z meters and you then decrease one side by 1 meter and increase one side by 1 meter then your total area decreases by 1 meter squared.

With this rule, I’m guessing that 99 x 101 is 9,999? Is that correct?

Me: 99 x 101.

QAMA: Give me an estimate.

Me (hoping to be exact): 9,999.

QAMA: 9,999.

Me (feeling): Yes, it seems that our rule works. Going from a plot of land of 100 meters by 100 meters (10,000 square meters), to a plot of land of 99 meters by 101 meters, decreases our land size by 1 meter squared.

But why is that? What is happening with 19 x 21, 39 x 41, 15 x 17 and 99 x 101? What is common to all the calculations above?

Let Z = 20, or 40 or 16, or 100. In each of these calculations we had:  (Z-1) x (Z+1).

For example when Z = 40, this is: (40-1) x (40+1). Now simple algebra yields,

(Z-1) x (Z+1) = Z x (Z+1) – 1 x (Z+1) = Z x Z + Z – Z – 1 = Z x Z - 1.

So for example, what is 89 x 91? Take here Z = 90 and get,

(90-1) x (90+1) = 90 x 90 – 1 = 8,099.

I think it is a very neat exploration, don't you? Let's try to take it even a bit further. What we had was one way for approximating W x V when the difference of W and V is 2. For example W = 19 and Z = 21. But what to do with general numbers? Say this case:

Me: 23 x 29

QAMA: Give me an estimate.

Me (thinking): OK, let’s take Z = 26 because it is in the middle of 23 and 29. Then our calculation is like (Z-3) x (Z+3). Then using algebra in a similar manner to what we did before,

(Z-3) x (Z+3) = Z x Z – 9.

Me (thinking more): So as an estimate I’ll just key in Z x Z.

Me (ohhh….): But 26 x 26 is still hard for me, so I’ll just go for 25 x 25. That I can probably work out in my head:

25 x 25 = (10+10+5) x 25=10 x 25 +10 x 25+5 x 25 = 250+250 + 125 = 625

Me (keying the result with some hesitation): 625.

QAMA: 667

Me (feeling): Accomplished! I knew to estimate that the area of a plot of land being 23 meters by 29 meters is around 625 meters. QAMA then gave me the exact result after accepting my estimate.

After using QAMA for a while I certainly suggest it for children, teenagers, teachers, parents, engineers, accountants and even grandparents. There is something very satisfying in having to think a bit before getting a precise answer. QAMA helps you stay on top of your game.

You can use QAMA more and more and as you do, you’ll see that things that you might have taken for granted in basic mathematics suddenly pop up and drive your curiosity. You may discover rules such as the ones we just discovered above. In my case, I will be using QAMA to nurture connection with my own children.

You can try the QAMA calculator on iOS, Android or as a hand held calculator on dedicated hardware. I recommend it.

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