# What is the square of 5 1

**What is a square number?**

A square number is a number that can be written as the product of two equal factors from natural numbers.

Formulas: k = a * a = a² (k and a stand for natural numbers.)

After that, a number like 4/9 = (2/3) ² is excluded here.

Conversely, a square number is created when you multiply a natural number by itself.

Formulas: a * a = a² = k (k and a stand for natural numbers.)

The same factor is called the base number.

These are the first 100 square numbers.

The name is explained by the illustration of the square numbers:

One finds out

**:**Only the unit digits 0, 1, 4, 5, 6 or 9 appear in the table.

Here is a consideration:

For example, you first pick out the three-digit numbers. They can be represented as 100c + 10b + a, where a, b and c are single-digit numbers.

We have (100c + 10b + a) ² = [(100c + 10b) + a] ² = (100c + 10b) ² + 2 (100c + 10b) a + a² = 100 (10c + b) ² + 10 (20c + 2b) a + a²

= 10 {[10 (10c + b) ²] + (20c + 2b) a} + a².

This means that the square of a three-digit number can be written in the form 10x + a². The one digit of the square is determined solely by a², and these are 0, 1, 4, 9 or the last digits of 16, 25, 36, 49, 64 and 81.

These considerations can be applied to all multi-digit square numbers.

**square root **Top

Finding a square number is easy. It is more difficult to find the base number for a square number.

This process is called "pulling the root" or square root.

Determining a square root is no longer a problem today.

You type the number into a calculator and press the root key: 17424 - root key - 132.

Further methods are discussed using the example sqrt (17424).

It is well known that instead of sqrt (17424) one also writes | ......................................................................... |

1 Determination via an interval nesting

The number must be between 100 and 200 (100² = 10000,200² = 40000)

The number is between 130 and 140 (130² = 16900 and 140² = 19600)

The ones digit of 17424 is 4. Then only 132 and 138 are possible.

The following applies: sqrt (17424) = 132.

2 Determination of a factorization

You split the number into factors and develop the square number from this.

17424 = 8 * 2178 = 16 * 1089 = 16 * 9 * 121. Then the square root is 4 * 3 * 11 = 132.

3 school method from bygone times

In terms of form, too, it is a kind of written division.

1

...... | Divide the number from the right into pairs of digits. |

2

Find the square number that is less than or equal to the pair on the left. This is trivially 01 here. Subtract the square number. The difference is 0. |

3

Get down the next pair of numbers similar to writing dividing. In an additional calculation, divide 074 by ten times the doubled number in the top right of the first line. That number is 20. The quotient is 3. The remainder in the division is of no interest. |

4

The ancillary calculation above is usually done in your head. Add the number 3 you found to the last divisor (20), and you get 23. Write the 3 of the additional calculation in the first line. |

5

As with the written division, form 3 * 23, write the product on the next line and subtract. It results in 5. |

6

Get the next pair down, 24. In an additional calculation, divide the resulting number 524 by ten times the double number 13 in the first line. That is 260. It results in 2, a remainder remains. |

7

Write the two on the first line on the right and add it to 260. Divide the number 524 below by 262 as you would when dividing in writing. Find the difference. It is zero. So sqrt (17424) = 132 applies. |

The explanation of this method results from the "trinomic formula" (a + b + c) ² = a² + b² + c² + 2ab + 2ac + 2bc.

...... | ...... | If one sets a = 100, b = 30 and c = 2, so you get the numerical example. |

Before calculators existed, this method was the standard method in school and was practiced extensively. Today it is still listed as a one-off in some textbooks.

(5, page 47 "Root extraction by hand")

4 Determination according to Newton's method (or Heron method)

In this method, the convergent sequence x

_{1}= 1 and x

_{k}= (x

_{k-1}+ n / x

_{k-1}) / 2 the root x = sqrt (n) is determined more and more precisely.

It may be interesting for the younger generation to see how the procedure was programmed and illustrated in the past. The predominant programming language in the 1970s was BASIC.

10 REM *** NEWTON'S APPROXIMATION 20 REM FOR DETERMINING THE SQUARE ROOT *** 30 INPUT Y, E 40 LET A = 1 50 LET B = Y / A 60 LET M = .5 * (A + B) 70 IF ABS (A-B) 90 GOTO 50 100 PRINT M 110 END |

You are amazed: only after eleven steps does the root 132 result.

The reason is the unfavorable initial value A = 1. It should be as close as possible to the root.

The "Heron Approach" is A = Y.

**consequences **Top

There are five infinite sequences associated with square numbers. These are

> the sequence of natural numbers,

> the sequence of the square numbers,

> the sequence of the partial sums of the square numbers, i.e. the series of square numbers,

> the sequence of the reciprocal values of the square numbers,

> the series of reciprocal values of the square numbers.

To the first and second episode

It cannot be overlooked that up to a certain number there are more natural numbers than square numbers. This statement is grasped in a first approximation to the problem by a way of speaking the set theory:

The set of square numbers is a real subset of the set of natural numbers.

To the second episode

The sequence of the square numbers is also known as the arithmetic sequence of the second order.

The formula for this is q

_{n}-q

_{n-1}= 2n-1.

To the third episode

s

_{1}=1²=1

s

_{2}=1²+2²=5

s

_{3}=1²+2²+3²=14

s

_{4}=1²+2²+3²+4²=30,

**...**

In general, s

_{n}= 1² + 2² + 3² + ... + n² = n (n + 1) (2n + 1) / 6.

Possible derivation

Approach: s_{n}= An³ + Bn² + Cn | s_{1}= 1 results in 1 = A + B + Cs _{2}= 5 gives 5 = 8A + 4B + Cs _{3}= 14 gives 14 = 27A + 9B + 3C | It follows A = 1/3, B = 1/2, C = 1/6 |

_{n}= (1/3) n³ + (1/2) n² + (1/6) n = n (n + 1) (2n + 1) / 6.

(4, page 28f.)

To the fourth and fifth episodes

The fourth episode is 1/1, 1/4, 1/9, 1/16,

**...**, 1 / n². It has the limit value 0 for n towards infinity.

The associated infinite series 1/1 + 1/4 + 1/9 + 1/16 + is interesting,

**...**

It is convergent and has the limit value pi² / 6 (according to Euler) or rounded 1.65.

(3, page 93f. And page 241)

The corresponding series formed from the prime numbers, namely "Sum of 1 / p", is divergent. From this one concludes that the prime numbers are closer than the square numbers (2, page 322).

**to hum **Top

Sum of the odd numbers

The sum of the first odd numbers is a square number. More precisely, 1 + 3 + 5 + ... + (2n-1) = n².

Example: 1 + 3 + 5 + 7 + 9 = 5².

Difference between two square numbers

Every prime number p> 2 can be clearly represented as the difference between two square numbers.

Example: 37 = 361-324 = 19²-18²

Division by 8

The squares of the odd numbers, when divided by 8, leave the remainder 1.

Example: 19²

**:**8=361

**:**8=45+1/8

Lagrange's theorem

Every natural number n can be represented as a sum of at most four square numbers.

Example: 85 = 64 + 16 + 4 + 1

Fermat-Euler's prime number theorem or Fermat's two-squares theorem

A prime number of the form 4n + 1 must be represented as the sum of two squares.

Examples: 5 = 1² + 2², 13 = 2² + 3², 625 = 7² + 24² = 15² + 20²

Pythagorean numbers

There are triples of numbers that satisfy the formula a² + b² = c². You can find out more about it on my website

__3-4-5 triangle__read more.

**Numbers games **Top

In this chapter, among other things, results from Lietzmann's book (7) from 1948 are compiled and supplemented with computer help.

Calculation from 51² to 59²

Example: 56² = 3136

(50 + E) ² = (25 + E) 100 + E²

Implementation: Add 25 to the ones digit and append the square of the ones digit.

Calculation of three-digit numbers with a 0 as a tens digit

Example: 203² = 4

**12**09, 609²=3

**708**81

(H0E) ² = H² * 10000 + 2U * 100 + E²

Implementation: The two spaces on the outside are formed by the squares of the hundreds digit and the ones digit. In the middle is the double product of hundreds and ones. If the double product has three digits, then the carryover 1 must be added to the square of the hundreds.

Squares from the digits 0 to 9. Each digit occurs exactly once.

There are 77 numbers.

1026 753 849 = 32043² is the smallest number.

9814072356 = 99066² is the largest number.

The ten-digit numbers from different digits are called "Pandigital numbers" in English.

Square numbers from the digits 1 to 9. Each digit occurs exactly once.

Two examples: 139845276 = 11826² and 923187456 = 30384²

The nine-digit numbers from the digits 1 to 9 are called "zeroless pandigital numbers" in English.

Mirror numbers

122 = 144 and 212 = 441 13² = 169 and 31² = 961 | 102² = 10404 and 201² = 40401 103² = 10609 and 301² = 90601 112² = 12544 and 211² = 44521 113² = 12769 and 3112 = 96721 | 1012² = 1024144 and 2101² = 4414201 1112² = 1236544 and 2111² = 4456321 1212² = 1468944 and 2121² = 4498641 2012² = 4048144 and 2102² = 4418404 |

Consecutive numbers

8281=91² | 183184=428² 328329=573² 528529=727² 715716=846² | 60996100=7810² 82428241=9079² 98029801=9901² | 1322413225=36365² 4049540496=63636² |

Palindromes under the square numbers

121=11² 484=22² 676=26² | 10201=101² 12321=111² 14641=121² 40804=202² 44944=212² 69696=264² 94249=307² | 698896=836² | 1002001=1001² 1234321=1111² 4008004=2002² 5221225=2285² 6948496=2636² | 123454321=11111² |

Five sequences of square numbers

1²=1 11²=121 111²=12321 1111²=1234321 ... | 4²=16 34²=1156 334²=111556 3334²=11115556 ... | 7²=49 67²=4489 667²=444889 6667²=44448889 ... | 9²=81 99²=9801 999²=998001 9999²=999800001 ... | 13²=169 133²=176 89 1333²=1 776889 13333²=1777 68889 ... |

Kaprekar numbers

A number like 703 is called the Kaprekar number because it has the following strange property:

If you square the number 703² = 494209 and divide the square into 494 and 209, the sum 494 + 209 is 703 again.

The Kaprekar numbers below 10000 are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777 and 9999.

**Figure numbers **Top

The square numbers belong to the figure numbers.

Square numbers Pentagonal numbers Hexagonal numbers Heptagon numbers Octagonal numbers ... | n² n * (3n-1) / 2 n * (4n-2) / 2 n * (5n-3) / 2 n * (3n-2) ... | 1 3 6 10 15 21 28... 1 4 9 16 25 36 49...1 5 12 22 35 51 70... 1 6 15 28 45 66 91... 1 7 18 34 55 81 112... 1 8 21 40 65 96 133...... |

At Mathworld you can find

Square numbers under the triangular numbers: 1, 36, 1225, 41616, 1413721, 48024900, ... Sloane's A001110

Square numbers among the pentagonal numbers: 1, 9801, 94109401, 903638458801, 8676736387298001, ... (Sloane's A036353)

More about square numbers can be found elsewhere on my homepage.

...

**Square numbers on the internet**Top

German

Andreas Göbel

Pulling a root - paper pencil method

Math prism

Square numbers, number acrobatics

Tino Hempel

The written drawing of a square root

Wikipedia

Square number, Square root, Centered square number, Root (mathematics), Written root extraction,

Four squares set

English

Eric W. Weisstein (MathWorld)

Square Number, Squarefree, Squareful, Antisquare Number, Square Triangular Number, Pentagonal Square Number,

Heptagonal Square Number, Octagonal Square Number, Square Root, Square Root Algorithms, Newton's Iteration

Patrick De Geest (World of Numbers)

Extraordinary squares and powers (non-palindromic allowed)

Neil J.A. Sloane

Integer sequences

The squares: A000290

Wikipedia

Square number, Polygonal_number, Triangular square number, Pandigital_number, 9814072356 (number),

Square root, Methods of computing square roots, Euler's four-square-identity, Kaprekar number,

Lagrange's four-square theorem

**credentials **Top

(1) Hermann Schubert: Mathematical leisure hours, Berlin 1941 (headword: "Ludendo discimus")

(2) Heinrich Behnke et al. (Ed.): Mathematik 1, DAS FISCHER LEXIKON, Frankfurt am Main 1964

(3) Jean-Paul Delahaye: Pi - the story, Basel 1999 [ISBN 3-7643-6056-9]

(4) Maximilan Miller: Solved and unsolved mathematical problems, Leipzig 1982

(5) August Schmid et al .: Lambacher / Schweizer LS9, Stuttgart 1996 [ISBN 3-12-730740-3]

(6) Forsythe, Keenan, Organick, Stenberg: Programming with BASIC, Braunschweig 1976 [ISBN 3-594-10854-6]

(7) Walter Lietzmann: Oddities in the realm of numbers, Bonn 1948

**Feedback:**Email address on my main page

This page is also available in German.

URL of my homepage:

http://www.mathematische-basteleien.de/

© 2005 Jürgen Köller

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