As I have promised, I will teach you more math tricks that will impress your friends. The most exciting part, however, is not actually the trick but why the trick works. In this post, we are going to learn math trick which we will call magic 1089, a trick I learned at BasicMathematics.com.
The 1089 Math Trick
Step 1: Think of a 3-digit number where its digits are decreasing.
Step 2: Reverse the order of the digits.
Step 3: Subtract the number in step 2 from the number in step 1.
Step 4: Reverse the order of the difference in step 3.
Step 5: Add the numbers in step 3 and step 4.
The result is 1089.
Let’s take 3 examples.
Step 1: Let us take 985.
Step 2: Let us reverse the digits making the number 589.
Step 3: Subtract the numbers: 985 – 589 = 396.
Step 4: Reverse the digits of the number in Step 3: 693.
Step 5: Add the numbers in Step 3 and 4: 396 + 693 = 1089.
Step 1: Let us take 742.
Step 2: Let us reverse the digits making the number 247.
Step 3: Subtract the numbers: 742 – 247 = 495.
Step 4: Reverse the digits of the number in Step 3: 594.
Step 5: Add the numbers in Step 3 and 4: 495 + 594 = 1089.
Step 1: Let us take 831.
Step 2: Let us reverse the digits making the number 138.
Step 3: Subtract the numbers: 831 – 138= 693.
Step 4: Reverse the digits of the number in Step 3: 396
Step 5: Add the numbers in Step 3 and 4: 693 + 396= 1089
Hmmm… the results in the three examples was always 1089.
What Really Happened
Before we discuss why the math trick works, let us observe what happened when we subtracted the digits of the numbers in step 3. Take note of these observations because they are the keys to the proof why the math trick works.
The condition states the digits of the number chosen in Step 1 is decreasing. Now, it follows that when digits of the number is reversed, the ones digit of the subtrahend is larger than the ones digit of the minuend. This means that we have to “borrow 1” from the tens digit (strictly, we are borrowing 10).
When a 3-digit number is reversed, the tens digit of the original number and the number with reversed digits are always the same. The tens digit of 985 is 8 and the tens digit of 589 is 8. In the subtraction, we have borrowed 1 from 8 when we subtracted the ones digit, so the tens digit of the subtrahend is greater than that of the minuend. In fact, this is always the case (Why?).
We also have to “borrow 10” from the hundreds digit and “add it” to the tens digit since we have borrowed 10 from the tens digit in Observation 1.
Why the Math Trick Works
We can generalize our observations above and use them to prove why the math trick works. Let us discuss the generalization of the subtraction per digit; that is, we subtract each pair of digits independently, so we do not really consider their place values in relation to the original number. In the generalization, let the abc be the 3-digit number with digits a, b, and c where a > b > c.
Ones: 10 + c – a
Since the number abc has decreasing digits, in subtracting ones digit, we always have to “borrow 1” from the tens digit since a > c. So, we add 10 to c and then subtract a from their sum. Therefore, the ones digit is 10 + c – a.
As we have mentioned in Observation 2, the tens digit of the original and the reversed numbers are the same. Since we borrowed 1 from the minuend, the subtrahend is always greater than 1. This makes the difference of the tens digit 9. This is shown in the generalized subtraction (b – 1 + 10) – b = 9 (Can you see why?).
Hundreds: a – 1 – c
We borrowed 1 from a, making it a – 1. Subtracting, we have a – 1 – c.
Reversing the Difference and Adding
Shown in the figure below is the sum of the difference of the numbers we subtracted and that of which its digits are reversed. The ones digit add up to 9, the tens add up to 18, “carrying 1” to the tens digit. The tens digit add up to 9 + 1 = 10. This makes the digits of the sum 10, 8, and 9 which is 1089.
There are several things we need to explore. What if the two digits of the three digit number we chose are the same, say, 533. Will the trick still work? What if the ones digit is 0, say 620. Will the trick still work.
Hmmm… This time, it’s your time to explore.