1089 (number)
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1089 is the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
after 1088 and before 1090. It is a
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
(33 squared), a
nonagonal number A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the constr ...
, a 32-gonal number, a 364-gonal number, and a
centered octagonal number A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the ...
. 1089 is the first reverse-divisible number. The next is 2178 , 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 > C; if A < 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 × ˆ’1 + 10+ 180 = 1089.


Other properties

Multiplying the number 1089 by the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 bases, e.g.
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
1067 or
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, h ...
10EF, also have these properties.


Extragalactic astronomy

The numerical value of the
cosmic microwave background radiation In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
redshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
is about ( corresponds to present time)


Other uses

* In the
Rich Text Format ) As an example, the following RTF code would be rendered as follows: This is some bold text. Character encoding A standard RTF file can only consist of 7-bit ASCII characters, but can use escape sequences to encode other characters. T ...
, the language code 1089 indicates the text is in
Swahili Swahili may refer to: * Swahili language, a Bantu language official in Kenya, Tanzania and Uganda and widely spoken in the African Great Lakes * Swahili people, an ethnic group in East Africa * Swahili culture Swahili culture is the culture of ...
.Microsoft, ''Microsoft Office Word 2007 Rich Text Format (RTF) Specification'' February (2007): 142. The hexadecimal number 441 (decimal 1089) is identified with "Kiswahili (Kenya)."


References

{{reflist Integers