Now let’s return to the original oddity of the number 1,089. We assumed that any number we chose would lead us to 1,089. Ask students how they can be sure. Well, they could try all possible three-digit numbers to see if it works. That would be tedious and not particularly elegant. An investigation of this oddity is within reach of a good elementary algebra student. So for the more ambitious students, who might be curious about this phenomenon, we will provide an algebraic explanation as to why it “works.”
We shall represent the arbitrarily selected three-digit number, htu, as
100h + 10t + u, where h represents the hundreds digit, t represents the tens digit, and u represents the units digit.
Let h > u, which would be the case in either the number you selected or the reverse of it. In the subtraction, u ?h < 0; therefore, take 1 from the tens place (of the minuend), making the units place 10 + u.
Since the tens digits of the two numbers to be subtracted are equal, and 1 was taken from the tens digit of the minuend, then the value of this digit is 10t ? 1. The hundreds digit of the minuend is h ? 1, because 1 was taken away to enable subtraction in the tens place, making the value of the tens digit 10t ? 1 + 100 = 10t + 9.
We can now do the first subtraction:
100h ? 1 +10t + 9+u + 10
100u +10t +h
100h ? u ? 1+109 +u ? h + 10
Reversing the digits of this difference gives us
100u ? h + 10 + 109 + h ? u ? 1
Now adding these last two expressions gives us
1009 + 1018 + 10 ? 1 = 1089
It is important to stress that algebra enables us to inspect the arithmetic process, regardless of the number.
Before we leave the number 1,089, we should point out to students that it has one other oddity, namely,
332 = 1089 = 652 ? 562
which is unique among two-digit numbers.
By this time your students must agree that there is a particular beauty in the number 1,089.
Taken From :Math Wonders to inspire teacher and student
3 responses so far ↓
1 kk raghuthaman // Dec 15, 2008 at 6:35 pm
11*99=1089
0101*9999= 01009899
001001*999999= 00100o998999
00010001*99999999= 0001000099989999
…..
And so on mental computing progresses
You will find start factors are mede one less and then each alternate digits sum is made 9. Vedic mathematics (original ancient Indian mathematics) could have used these and so reviver of this system of computing called it Vedic Mathematics. Simplest virtue of it is a matrix by matrix 2D square matrix positions relating which almost any body can do mentally. In a manner I have written answers you may try and verify answers in a computer.
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