Continuing with my Maths posts, let me deal with one more easy way to Maths. We all love 'Five', may be because we have 'five' fingers and 'five' toes in each of our limbs; well mostly. But '5' is a lot more useful in Mathematics. It is a wonderful number that makes Maths lot more easier, fun and even entertaining.
I asked my daughter, "Do you know the easy way to obtain whole squares of numbers ending with 5 ? Like say, 25, 35, 45 or even 115 and so on ?" I knew she doesn't and hence went on to explain. This isn't anything new again, but somehow not many Maths teachers follow this and I don't know why !
Let me first show how we do it and then explain how it actually happens. We have to remember all numbers ending with 5 will end with 25 when squared. The 25 will remain constant. We just have to look at the first number. Let us say 'u' and multiply that with (u+1) and the product has to be written before 25 and we have a whole square.
u x (u + 1) before 25
Let us take 25: u here is 2. Hence: 2 X 3 = 6. The whole square of 25 is 625.
Let us take 35: u here is 3. Hence: 3 x 4 = 12. The whole square of 35 is 1225.
Let us take 45: u here is 4. Hence: 4 x 5 = 20. The whole square of 45 is 2025.
Let us take 55: u here is 5. Hence: 5 x 6 = 30. The whole square of 55 is 3025.
This will go on and on. Now let us see how this actually happens. Once we know that, we will never have a problem with 'Five' !
-> Let us begin with 55. What is the easiest way to square a big number ?
-> We have already dealt with that in (a + b)2= a2 + 2ab + b2.
-> When we deal with any number ending from 5, we know the 'b' in the equation and that is 5.
-> So the equation gets simplified as follows.
* a2 + (2 x a x 5) + 25.
-> That is further simplified into
* a2 + 10 x a + 25
-> Now what are we left with ?
* a2 + 10a + 25
Now let us keep the 25 away for the time being because we are going to add it at the end. Thus we have
* a2 + 10a
This is further simplified into
* a (a +10)
-> In reality 'a' is a round figure here like 10, 20, 30, 40, 50, so on to 90, 100, 110 and more. Multiplying a round figure with a round figure [a x (a + 10)] always yields a multiple of 100 with 00 at the end. We just have to ignore those zeros for the sake of ease and speed. Thus the 00 goes with 25 as just 25.
So we have to remove the '0's from [a] and [a + 10] to get the smaller simpler numbers.
-> Thus we have:
*a / 10 and [a + 10] / 10
Thus the equation becomes:
* (a / 10) x (a + 10) / 10 = (a / 10) x (a / 10 + 1)
-> [a / 10] here is the simplified small number after removing the 'Zero'.
So we will name it [u] for convenience. So the equation becomes simpler:
* u x (u + 1) with the 00 going with 25.
Now we just put the product of u x (u + 1) before 25 and we have the whole square.
Let us take 145: u here is 14. Hence: 14 x 15 = 210. The whole square of 145 is 21025 !
Simply put, what are we doing ?
Take the number before 5 and multiply that with the one more than the same number and put the product before 25. We are home !
Hope I have made it simpler for those who like to make things easier in life; especially Mathematics !