Today we were playing cricket on top of our building. The game has rules, it’s not like playing in a ground with a lot of freedom to hit the ball, it’s about hitting the ball within the boundaries of the building, you’re out if the ball goes out without bouncing inside, there is a problem as well, if ball goes out side it may hit the costly window glasses of our neighbors, cars, some times to the people.. Another condition is, you are out if the fielder takes direct catch or one bounce catch with one hand. So the game is very tough to survive for long scores.

What ever the conditions, If someone follows the rules of physics, he can play well. I understood that very well and I hit only the loose deliveries not otherwise. But one cannot always be successful in accurately calculating the ball’s speed, where it is pitching in order to decide how to hit the ball. Some times it happens with me. Today we brought the new ball, as it is new it is bouncing than normal. We were played few matches, used the bounce to others out quickly. It’s my turn now, I know where the ball pitches and I know my shot to gain maximum out of it. But a little more power makes the ball go out side the boundaries and what? I’m out.

We started searching for the ball, it was not in our vicinity. Now the actual conversation began. It’s common to us that if ball falls into our neighbors house, we don’t ask them because they don’t give us. “Then my team-mate started calculating how many balls we lost in the way, and how much each is. The calculation was like 10 balls and each of  25 Rupees. So 5 x 10 = 50 & 20 x 10 = 200, the total is 250”, he said.

I wondered and asked him that did you really do that for such simple calculation.I must admit may be it’s his perspective to make things easy. Another thing is he broke the problem into chunks , solved the pieces, and joined them to arrive at answer. It’s really good for many other strategies as well in the real life problems. If you are an Engineer, you might have come across the Fast Fourier Transform(FFT), it uses the same technique. I apply the strategy anywhere I feel, to solve a problem with complexity. For instance, I use it to multiply numbers. Let 14 x 8, I know it is simple but there is a point to make with it, you can rewrite it as

(10+4) x 8 => 10 x 8 + 4 x 8 => 80 + 32 = 112

or

(10+4) x (10 – 2) = (100 -20) + (40 -8) = 80 + 32 = 112

Again the same. The approach you follow may vary in correlation with your thinking and learning process you had. One can further decompose the numbers more and more to make it much easier.

After all, just after having dinner, I asked my friend – ” What is the value of 18 x 17 “. Don’t bother weather answered or not but I got a sudden epiphany. I saw it from a Math student perspective. I saw perfect squares in them. One is 17^2 and the other is 18^2.

18 x 17

(17+1) x 17

17^2 + 17

if you know 17^2 = 289 that’s okay add 17 more to it. That is you answer

289+ 17 = 306

or you can further decompose the

17^2 as (10+7)^2 = 10^2 + 7^2 + 2 x 10 x 7 = 289

289 + 17 = 306

On the other hand, we can go for this

18 x 17

18 x (18 -1)

18^2 – 18

Same as above you will get the answer after subtracting 18 from its square, i.e., 306.

Aha, did you observe that, we see that both are same. 17^2 + 17 = 18^2 – 18 

Can we generalize them to see what happening exactly. Lets see, consider ‘n’ is a number, therefore

n * (n+1)       ||        (n+1) * (n+1 -1)

n*(n+1) = (n+1)*n

This is very simple. Lets do something more interesting

17^2 + 17 = 18^2 – 18 

let x = 17 , then

x^2 + x = (x+1)^2 – (x+1)

(x+1)^2 – x^2 = (x+1) +x

Did you see anything that we learnt in the six grade algebra lessons. let x+1 be a, and x be b, therefore,

a^2 – b^2 = a+b

(a+b)(a-b) = a+b

if and only if a and b are consecutive numbers. if a>b then(a-b) =1 and (a+b) takes positive sign, if b>a,then (a-b) = -1 and (a+b) takes a negative sign.

On the whole it’s all about how we see at problems. The more chunks you make, the more easy the problem to solve. Remember the divide and rule policy. I have applied it for many problems in my engineering and real life. The way of thinking varies from person to person, So what is your perspective?

Share your Ideas in comment section. Have fun with math.

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6 thoughts on “A Simple Multiplication : Mathematician’s Perspective

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