1089 (number)

1089 1, 3, 9, 11, 33, 99, 121, 363, 1089 1089 is the integer and natural number following 1088 and preceding 1090. In mathematics It is a square number (33 squared), a nonagonal number,Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers https://oeis.org/A001106 The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-06-02 a 32-gonal number, a 364-gonal number, and a centered octagonal number.Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers https://oeis.org/A016754 The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-06-02 1089 is the first nontrivial decimal reverse-divisible number: multiplying 1089 by 9 gives 9801, the reversal of its digits.A001232 Numbers n such that 9*n = (n written backwards) The next is 2178 (= 1089 × 2 = 8712/4), and they are the only four-digit numbers (disregarding palindromes) 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.). 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 = 1089.1089 and a Property of 3-digit Numbers http://www.cut-the-knot.org/Curriculum/Arithmetic/S1089.shtml 28 May 2015 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) Other uses In the Rich Text Format, the language code 1089 indicates the text is in Swahili.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 10 Category:Integers