Mirror number

A mirror number (sometimes also: invert number , inverse number or reciprocal number ) for a multi-digit natural number can be obtained by writing the digits in reverse order, e.g. B. 4321 is the mirror number of 1234. A number without a mirror number ends with the digit 0, e.g. B. 1230 in reverse order is 0321 = 321, only three digits.

If the same number results when inverting a number, it is called a number palindrome .
Even the sum of two mirror numbers always results in a palindrome if the sum of the digits at each digit remains less than ten, i.e. there is no number transfer in the written addition , which destroys the symmetry of the result.
But even if you add their mirror number to the sum of a mirror number pair, a palindromic number results, usually after a few steps, e.g. B. 39 + 93 = 132 and 132 + 231 = 363. With 89 + 98 24 steps are necessary; only with a few exceptions, the Lychrel numbers , does this algorithm not work.
Special mirror numbers are mirp numbers , i. H. Prime numbers which, read backwards, result in a prime number.
The difference between a number and its mirror number is (in the decimal system ) divisible by 9 (or a multiple of 9).
The multiplication of a number by its mirror number is particularly easy in mental arithmetic .
Mirror numbers of square numbers of some natural numbers behave like their squared mirror number, e.g. B .:

12²     = 144          |          441 = 21²
13²     = 169          |          961 = 31²
112²    = 12544        |        44521 = 211²
113²    = 12769        |        96721 = 311²
1112²   = 1236544      |      4456321 = 2111²
1113²   = 1238769      |      9678321 = 3111²
11112²  = 123476544    |    445674321 = 21111²
11113³  = 123498769    |    967894321 = 31111²
111112² = 12345876544  |  44567854321 = 211111²
1111112²= 1234569876544|4456789654321 = 2111111²

For 11, 111 etc. there are palindromic numbers (see table there).

Occurrence

Mirror numbers appear in mathematics didactics in arithmetic exercises, in tasks in mathematics competitions, in programming exercises for beginners, in some algorithms (such as the calculation of Kaprekar constants ) and in numerology .

literature

Kröber, KG Mathematics of Palindromes. Rowohlt 2003. ISBN 9783499615764

Individual evidence

↑ [1] http://www.jasondoucette.com/pal/89 , also enter other numbers and calculate up to the palindrome, accessed on May 4th
↑ Archived copy ( memento of the original dated July 31, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. http://www.helmholtz-bi.de/ . Retrieved July 31, 2016. @1@ 2

↑ [2] Mirror number arithmetic problems. Retrieved June 13, 2014.

↑ [3] http://www.programmingsimplified.com/c/source-code/c-program-reverse-number , C program for calculating mirror numbers (reverse number). Retrieved July 31, 2016.

[1] [1] http://www.jasondoucette.com/pal/89 , also enter other numbers and calculate up to the palindrome, accessed on May 4th

[2] Archived copy ( memento of the original dated July 31, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. http://www.helmholtz-bi.de/ . Retrieved July 31, 2016. @1@ 2

[3] [2] Mirror number arithmetic problems. Retrieved June 13, 2014.

[4] [3] http://www.programmingsimplified.com/c/source-code/c-program-reverse-number , C program for calculating mirror numbers (reverse number). Retrieved July 31, 2016.