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{{infobox number
| number        = 1089
| divisor       = 1, 3, 9, 11, 33, 99, 121, 363, 1089
}}

'''1089''' is the [[Integer|integer]] after 1088 and before 1090. It is a [[Square number|square number]] (33 squared), a nonagonal number,<ref>{{cite web|title=Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers|url=https://oeis.org/A001106|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|accessdate=2016-06-02}}</ref> a 32-gonal number, a 364-gonal number, and a centered octagonal number.<ref>{{cite web|title=Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers|url=https://oeis.org/A016754|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|accessdate=2016-06-02}}</ref> 1089 is the first reverse-divisible number. The next is 2178  {{nowrap|1=(= 1089 × 2 = 8712/4)}}, and they are the only four-digit numbers that divide their reverse.

==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 maths 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. 

In base 10, the following steps always yield 1089:
# Take any three-digit number where the first and last digits differ by more than 1.
# 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'''

as expected. On the other hand, if the spectator chooses 102 (or 201):

: '''201''' − '''102''' = '''99'''
: '''99'''  + '''99'''  ≠ '''1089'''

contradicting the rule. However, if we amend the third rule by reading '''99''' as a three-digit number '''099''' and take its reverse, we obtain:

: '''201''' − '''102''' = '''''099'''''
: '''''099''''' + '''990''' = '''1089'''

as expected.

===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.  Their difference is 99 × ('''A''' − '''C''') (For convenience, we assume A &gt; C; if A &lt; C, we first swap A and C.). Note that if '''A''' − '''C''' is 0, the difference is 0, and we do not get a 3-digit number for the next step. If '''A''' − '''C''' is 1, the difference is 99. Using a leading 0 gives us 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'''.<ref>{{cite web|title=1089 and a Property of 3-digit Numbers|url=http://www.cut-the-knot.org/Curriculum/Arithmetic/S1089.shtml|accessdate=28 May 2015}}</ref>

===Other properties===
Multiplying the number 1089 by the [[Integer|integer]]s 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 [[Radix|base]]s, e.g. [[Octal|octal]] 1067 or [[Hexadecimal|hexadecimal]] 10EF, also have these properties.

== Extragalactic astronomy ==
The numerical value of the cosmic microwave background radiation [[Astronomy:Redshift|redshift]] is about {{nowrap|1=''z'' = 1089}} ({{nowrap|1=''z'' = 0}} corresponds to present time)

==Other uses==
* In the [[Rich Text Format]], the language code 1089 indicates the text is in [[Social:Swahili language|Swahili]].<ref>Microsoft, ''Microsoft Office Word 2007 Rich Text Format (RTF) Specification'' February (2007): 142. The hexadecimal number 441 (decimal 1089) is identified with "Kiswahili (Kenya)."</ref>

==References==
{{reflist}}

[[Category:Integers]]

{{Sourceattribution|1089 (number)}}