1089 (number)
| ||||
---|---|---|---|---|
Cardinal | one thousand eighty-nine | |||
Ordinal | 1089th (one thousand eighty-ninth) | |||
Factorization | 32 × 112 | |||
Divisors | 1, 3, 9, 11, 33, 99, 121, 363, 1089 | |||
Greek numeral | ,ΑΠΘ´ | |||
Roman numeral | MLXXXIX | |||
Binary | 100010000012 | |||
Ternary | 11111003 | |||
Senary | 50136 | |||
Octal | 21018 | |||
Duodecimal | 76912 | |||
Hexadecimal | 44116 |
1089 is the
In magic
1089 is widely used in
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.[3]
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
Other uses
- In the Rich Text Format, the language code 1089 indicates the text is in Swahili.[4]
References
- ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ "Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ "1089 and a Property of 3-digit Numbers". Retrieved 28 May 2015.
- ^ Microsoft, Microsoft Office Word 2007 Rich Text Format (RTF) Specification February (2007): 142. The hexadecimal number 441 (decimal 1089) is identified with "Kiswahili (Kenya)."