← 1088 1089 Factorization 3× 11 | 1089 1090 → | |

Cardinal one thousand eighty-nine Ordinal 1089th (one thousand and eighty-ninth) Divisors 1, 3, 9, 11, 33, 99, 121, 363, 1089 |

**1089** is the integer after 1088 and before 1090. It is a square number (33 squared), a nonagonal number, a 32-gonal number, a 364-gonal number, and a centered octagonal number. 1089 is the first reverse-divisible number. The next is 2178 (= 1089 × 2 = 8712/4), and they are the only four-digit numbers that divide their reverse.

## Contents

## In magic

1089 is widely used in magic tricks because it can be "produced" from any two three-digit numbers. This allows it to be used as the basis for a Magician's Choice. For instance, one variation of the book test starts by having the spectator choose any two suitable numbers and then apply some basic math to produce a single four-digit number. That number is always 1089. The spectator is then asked to turn to page 108 of a book and read the 9th word, which the magician has memorized. To the audience it looks like the number is random, but through manipulation, the result is always the same. It is this property that led University of Oxford mathematician David Acheson to title his 2010 book '1089 and all that: a journey into mathematics'.

In base 10, the following steps always yield 1089:

- Take any three-digit number where the first and last digits differ by 2 or more.
- Reverse the digits, and subtract the smaller from the larger one.
- Add to this result the number produced by reversing its digits.

For example, if the spectator chooses 237 (or 732):

**732**−

**237**=

**495**

**495**+

**594**=

**1089**

## Explanation

The spectator's 3-digit number can be written as 100 × **A** + 10 × **B** + 1 × **C**, and its reversal as 100 × **C** + 10 × **B** + 1 × **A**, where 1 ≤ A ≤ 9, 0 ≤ B ≤ 9 and 1 ≤ C ≤ 9. (For convenience, we assume A > C; if A < C, we first swap A and C.) Their difference is 99 × (**A** − **C**). Note that if **A** − **C** is 0 or 1, the difference is 0 or 99, respectively, and we do not get a 3-digit number for the next step.

99 × (A − C) can also be written as 99 × [(A − C) − 1] + 99 = 100 × [(A − C) − 1] − 1 × [(A − C) − 1] + 90 + 9 = 100 × [(A − C) − 1] + 90 + 9 − (A − C) + 1 = 100 × [**(A − C) − 1**] + 10 × **9** + 1 × [**10 − (A − C)**]. (The first digit is **(A − C) − 1**, the second is **9** and the third is **10 − (A − C)**. As 2 ≤ A − C ≤ 9, both the first and third digits are guaranteed to be single digits.)

Its reversal is 100 × [**10 − (A − C)**] + 10 × **9** + 1 × [**(A − C) − 1**]. The sum is thus 101 × [**(A − C) − 1**] + 20 × **9** + 101 × [**10 − (A − C)**] = 101 × [**(A − C) − 1** + **10 − (A − C)**] + 20 × **9** = 101 × [−1 + 10] + 180 = **1089**.

## Other properties

Multiplying the number 1089 by the integers from 1 to 9 produces a pattern: multipliers adding up to 10 give products that are the digit reversals of each other:

1 × 1089 =**1089**↔ 9 × 1089 =

**9801**2 × 1089 =

**2178**↔ 8 × 1089 =

**8712**3 × 1089 =

**3267**↔ 7 × 1089 =

**7623**4 × 1089 =

**4356**↔ 6 × 1089 =

**6534**5 × 1089 =

**5445**↔ 5 × 1089 =

**5445**

Also note the patterns within each column:

1 × 1089 =**1089**2 × 1089 =

**2178**3 × 1089 =

**3267**4 × 1089 =

**4356**5 × 1089 =

**5445**6 × 1089 =

**6534**7 × 1089 =

**7623**8 × 1089 =

**8712**9 × 1089 =

**9801**

Numbers formed analogously in other bases, e.g. octal 1067 or hexadecimal 10EF, also have these properties.

## Extragalactic astronomy

The numerical value of the cosmic microwave background radiation redshift is about *z* = 1089 (*z* = 0 corresponds to present time)