I believe that mathematical reality lies outside of us. Our function is to discover, or observe it, and that the theorems which we describe grandiloquently as our “creations” are simply notes on our observations.
Godfrey H. Hardy
Numbers have a stunning beauty and there is a whole world of wonder to explore within them. This article focuses on a specific number that holds a special place in the name of this website. Let’s delve into the mathematical wonders of this number.
1089 is a unique number that possesses several fascinating mathematical properties. In this article, we will delve into the unique qualities of the number 1089 and examine its various properties.
1089 is the integer after 1088 and before 1090. It is a perfect square number and 332 = 1089.
When translated into binary form, 1089 is represented as 100010001. In Roman numerals, it is denoted as MLXXXIX.
Using the digits 1 to 9 in ascending and descending order
1089 = 12 × 34 + 5 × 6 + 78 + 9
1089 = 987 + 65 + 4 + 32 + 1
Using the digits 1, 7, 2 and 9
1089 = [17 + 2 × (9 + 17) × 2] × 9
Using the digits 9, 8, 7, 6 and 5
1089 = 9 × (8 × 7 + 65)
Using the digits 4, 3, 2, 1 and 0
1089 = (4! + 32)1 + 0!
Single digit representations
1089 is a frequently used number in magic tricks because it can be obtained from any two three-digit numbers. This feature makes it the basis for a magician’s choice. Start with a three-digit number and perform some basic arithmetic operations, the result is always 1089!
Magic. The steps to obtain 1089 from any two three-digit numbers are as follows:
∎ Choose any three-digit number whose first and last digits differ by 2 or more.
∎ Then reverse the digits, and subtract the smaller from the larger one.
∎ Finally, add the reversed number with the original one to obtain 1089.
Example 1. Consider the number 731 (difference between the first and last digits is 6 > 2).
Reversed number is 137.
Their difference is 731 – 137 = 594.
The reverse of the number 594 is 495.
The sum of the last two numbers is 594 + 495
= 1089.
Example 2. Consider the number 684 (difference between the first and last digits is 2).
Reversed number is 486.
Their difference is 684 – 486 = 198.
The reverse of the number 198 is 891.
The sum of the last two numbers is 198 + 891
= 1089.
Example 3. Consider the number 685 (difference between the first and last digits is 1 < 2).
Reversed number is 586.
Their difference is 685 – 586 = 99.
The reverse of the number 99 is 99.
The sum of the last two numbers is 99 + 99
= 198.
An algebraic proof of the trick
Let one of the 3-digit numbers be represented as ABC. The reversed number will then be CAB. Without loss of generality, let ABC be the larger of the two numbers, so that A > C (note that we did not allow A = C in the problem statement). We are considering numbers in base 10.
Represent ABC and CAB as 100A + 10B + C and 100C + 10B + A, respectively. Then, ABC – CAB is
ABC – CBA
= (100A + 10B + C) – (100C + 10B + A)
= 99A – 99C
= 99(A – C).
Since the first and last digits in ABC differ by at least 2, (A – C) must be any of 2, 3, 4, 5, 6, 7, 8, or 9. Therefore, 99(A – C) is one of the following:
- 99 × 2 = 198
- 99 × 3 = 297
- 99 × 4 = 396
- 99 × 5 = 495
- 99 × 6 = 594
- 99 × 7 = 693
- 99 × 8 = 792
- 99 × 9 = 891.
We want to add this number to its reverse. Let’s call the number XYZ, which is 100X + 10Y + Z, and ZYX is 100Z + 10Y + X. We want to add XYZ to ZYX.
Looking at the possibilities for XYZ above, we notice that the middle number Y is always 9. Also note that the first and third digits always add up to 9, i.e., X + Z = 9. Therefore,
XYZ + ZYX
= (100X + 10Y + Z) + (100Z + 10Y + X)
= 101X + 101Z + 20Y
= 101(X + Z) + 20Y
= 101(9) + 20(9)
= 909 + 180
= 1089, as expected.
The products formed by multiplying 1089 by integers from 1 to 9 exhibit a unique symmetry in the digits of the products. Some of the observations of this symmetry include:
- The thousand’s digits of the products form an increasing sequence from 1 to 9;
- The unit’s digits of the products form a decreasing sequence from 9 to 1, which is the reverse of the thousand’s digits;
- The sum of the digits in the unit’s place and the thousand’s place is always equal to 10;
- The hundred’s digits of the products form an increasing sequence from 0 to 8;
- The ten’s digits of the products form a decreasing sequence from 8 to 0, which is the reverse of the hundred’s digits.
- The sum of the digits in the tens’ place and the hundred’s place is always equal to 8.
- The sum of the digits in each product is always equal to 18, which is the sum of the digits in 1089.
When 1089 is multiplied by integers from 1 to 9, the products formed by multipliers that add up to 10 are digit reversals of each other. For example:
This pattern holds true for all multipliers that add up to 10, making 1089 a special number with unique properties. In particular, the multiplication of 1089 by 9 is of special interest. The digits are reversed to the original number as a result. In other words, 1089 turned into 9801.
The number 1089 has yet another cute property that depends upon its digits. The process is as follows:
- Consider a number;
- Calculate the sum of the squares of its digits;
- If it is 1 or 89, stop; otherwise
- Repeat the process until it reaches either 1 or 89.
Here are a few examples:
Example 1. Consider n = 1.
1 → 12 = 1 → 12 = 1 → . . .
It’s clear that once 1 is reached for a second time, the process enters a repeating loop and will always result in 1.
Example 2. Consider n = 28.
28 → 22 + 82 = 68 → 62 + 82 = 100 → 12 + 02 + 02 = 1 → . . .
Example 3. Consider n = 89.
89 → 82 + 92 = 145 → 12 + 42 + 52 = 42 → 42 + 22 = 20 → 22 + 02 = 4 → 42 = 16 → 12 + 62 = 37 → 32 + 72 = 58 → 52 + 82 = 89 → 82 + 92 = 145 → 12 + 42 + 52 = 42 → 42 + 22 = 20 → 22 + 02 = 4 → 42 = 16 → 12 + 62 = 37 → 32 + 72 = 58 → 52 + 82 = 89 → . . .
It’s clear that once 89 is reached for a second time, the process enters a repeating loop and will always result in 89.
Example 4. Consider n = 30.
30 → 32 + 02 = 9 → 92 = 81 → 82 + 12 = 65 → 62 + 52 = 61 → 62 + 12 = 37 → 32 + 72 = 58 → 52 + 82 = 89 → . . .
A two-digit number ab with a > b can be used to form a square using the following process:
∎∎ Choose any two-digit number ab with a > b;
∎ Reverse the digits to form ba;
∎ Finally, calculate the difference between the squares of ab and ba.
The only time this difference is a square is when the result is 1089.
In other words, 1089 is the only square that can be obtained from the difference between the squares of two two-digit numbers formed by reversing the digits of each other.
Proof. The algebraic explanation behind the above process is straightforward. The difference between the squares of two two-digit numbers ab = 10a + b and ba = 10b + a (obviously, 10 > a > b > 0) is
(10a + b)2 − (10b + a)2
= 100a2 + 20ab + b2 – 100b2 – 20ab – a2
= 99a2 – 99b2
= 11(a + b) ⋅ 9(a − b).
Clearly, this expression will be a square if and only if 11(a + b)(a − b) is a square. For this to happen, we must have 11 | (a + b)(a − b).
Since a > b, we get 11 ∤ (a − b). The only possibility is 11 ∣ (a + b). As 11 is a prime and 1 ≤ a + b ≤ 17, it follows that a + b = 11.
Thus, a – b should be a perfect square. We can write, a – b = (a + b) – 2b, so that when a – b is even (or odd), a + b is even (or odd). Therefore, a + b and a – b are of the same parity. Only two possibilities are thus a − b = 12 or a − b = 32.
If we take a − b = 32, then a = b + 9 giving the minimum value of a as 10, which is not possible. Thus, a = b + 1 giving a – b = 1.
Solving the two equations, we get a = 6 and b = 5.
More significantly, we have
652 – 562
= 112 ⋅ 32
= 332
= 1089.
An important observation:
Here is a magic square with magic sum 1089.
Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.