6174

Last updated • 3 min readFrom Wikipedia, The Free Encyclopedia
← 617361746175 →
Cardinal six thousand one hundred seventy-four
Ordinal 6174th
(six thousand one hundred seventy-fourth)
Factorization 2 × 32 × 73
Divisors 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174
Greek numeral ,ϚΡΟΔ´
Roman numeral VMCLXXIV, or VICLXXIV
Binary 11000000111102
Ternary 221102003
Senary 443306
Octal 140368
Duodecimal 36A612
Hexadecimal 181E16

6174 (six thousand, one hundred [and] seventy-four) is the natural number following 6173 and preceding 6175.

Contents

Kaprekar's constant

The natural integer 6174 is known as Kaprekar's constant, [1] [2] [3] after the Indian mathematician D. R. Kaprekar. This number is notable for the following curious behavior:

  1. Select any four-digit number which has at least two different digits (leading zeros are allowed),
  2. Create two new four-digit numbers by arranging the original digits in a.) ascending and b.) descending order (adding leading zeros if necessary).
  3. Subtract the smaller number from the bigger number.
  4. If the result is not 6174, return to step 2 and repeat.

This process, known as Kaprekar's routine, is guaranteed to reach a fixed point at the value 6174 in no more than 7 iterations, [4] at which point it will continue yielding that value (7641 - 1467 = 6174).

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4. For numbers with three identical digits and a fourth digit that is one higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading zero; for example: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 = 8082; 8820 – 288 = 8532; 8532 – 2358 = 6174. [5]

Other "Kaprekar's constants"

There can be analogous fixed points for digit lengths other than four; for instance, if we use 3-digit numbers, then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Kaprekar constants".

Applications

Convergence analysis

In numerical analysis, Kaprekar's constant can be used to analyze the convergence of a variety numerical methods. Numerical methods are used in engineering, various forms of calculus, coding, and many other mathematical and scientific fields.

Recursion theory

The properties of Kaprekar's routine allows for the study of recursive functions, ones which repeat previous values and generating sequences based on these values. Kaprekar's routine is a recursive arithmetic sequence, so it helps study the properties of recursive functions. [6]

Other properties

References

  1. ↑ Nishiyama, Yutaka (March 2006). "Mysterious number 6174". Plus Magazine .
  2. ↑ Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica . 15: 244–245.
  3. ↑ Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.
  4. ↑ Hanover 2017, p. 1, Overview.
  5. ↑ "Kaprekar's Iterations and Numbers". www.cut-the-knot.org. Retrieved 2022-09-21.
  6. ↑ "Kaprekar's Constant 6174: A Fascinating Mathematical Phenomenon".
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.