Square Numbers 
Contents of this page
What is a Square Number?
Square Root
Playing with Numbers
Figurate Numbers
Square Numbers on the Internet
To the main page   "Mathematische Basteleien"

What is a Square Number?
A square number is a number you can write as a product of two equal factors of natural numbers.
Formula: k=a*a=a² (k and a stand for integers.)

On the other hand a square number results by multiplying an integer by itself.
Formula: a*a=a²=k (a and k stand for integers.)
The same factor is called the base. 

After this a fractal number like the 4/9=(2/3)² is suspended.

These are the first 100 square numbers. 

You can illustrate square numbers by the following drawings. 

You can see: Only the digits 0, 1, 4, 5, 6 or 9 are on the ones place.
Thoughts concerning this point:
Consider numbers with three digits. They can be written as 100c+10b+a. The letters a, b, and c stand for digits.
There is (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. 
That means that the square of a three-digit number has the form 10x+a². Therefore the digit on the ones place is given only by a², and these are  0, 1, 4, 9 or the last digits of 16, 25, 36, 49, 64 und 81.

These principles can be transferred to all multi-digit numbers.

Square Root   top
It is easy to find a square number. It is more difficult to find the base of a square number. 
This procedure is called the extraction of the square root.

132²=17424 leads to the square root sqrt(17424)=132. 
It can also be written as .........................................................................

Today there is no problem to find square roots. There are calculators.
You put in the number and press the root key:  17424 -- root key -- 132. 
Result:  sqrt(17424)=132

More methods follow shown by the example.
1 Determination by nested intervals

The number must be between 100 and 200  (100²=10000 and 200²=40000).
It must be between 130 and 140 (130²=16900 and 140²=19600).
The 4 of 17424 is in the ones place. Then 132 and 138 come into consideration.
Result:  sqrt(17424)=132

2 Determination by factorization
You write the number in factors and develop the square number this way.
17424  =  8*2178  =  16*1089  =  16*9*121. Thus the root is 4*3*11=132.
Result: sqrt(17424)=132

3 School method from former times
This is a kind of long division.
...... Portion the number in digit pairs starting on the right. 

Look for a square number, which is equal or just beneath the left pair. This is 1, of course. Subtract the square number 1. You get 0. 

Bring down the next pair similar to the long division. Divide the tenfold of the double of the number, which stands in the first line on the right, in an extra calculation. This is 20. The quotient is 3. The remainder doesn't matter. 
Usually you do the extra calculation by heart. 

Add the found number 3 to the last divisor (20). You get 23. 
Write the number 3 of the extra calculation in the first line on the right.

...... Multiply 3*23 as in a long division, write the result 69 in the next line and subtract. 
You get 5. 

Bring down the next pair 24. Divide the number 524 by 260. 260 is the tenfold of the double of the number 13 in the first line. This is 260. You get 2. 
There is a remainder. 

...... Write the number 2 from the last step in the first line on the right. Add it to 260 and you get 262.
Divide 524 by 262 as at the long division. Subtract and you get 0.

This gives sqrt(17424)=132.

The explanation of this method results from the "trinomic formula"  (a+b+c)²=a²+b²+c²+2ab+2ac+2bc.
...... ...... Replace a=100, b=30 and c=2, 

so you get the number example above. 

Before the age of calculators this method was standard in German schools and was practised a lot. 
Today this is history. 
(5, page 47 "Wurzelziehen per Hand")

4 Computing the square root of a positive number by Newton's method
The square root of 17.424 will be found by the convergent sequence  x1=1 and xk=(xk-1+n/xk-1)/2 for k>1.

Maybe it is interesting for younger people to see how this method was programmed and illustrated in former times. The common programming language was BASIC in the 1970s. 
40 LET A=1
50 LET B=Y/A
60 LET M=.5*(A+B)
70 IF ABS(A-B)<E THEN 100
80 LET A=M
90 GOTO 50
110 END
(6, page 92f. )
It is surprising that the square root 132 is not found earlier than after ten steps. 
The reason is the bad start number A=1. It is better, if near the root. 
The Heron ansatz is A=Y.

Sequences   top
There are five sequences in connection to square numbers. 
>the sequence of the natural numbers
>the sequence of the square numbers
>the sequence of the partial sums of the square numbers, in other words the series of square numbers
>the sequence of the reciprocals of the square numbers
>the series of the reciprocals of the square numbers

To the first and second sequences
Looking at both sequences you could think that the amount of the squares is smaller than the amount of the natural numbers. This is wrong. There is no number "infinite" of infinite sequences.  You can rather say that natural numbers and square numbers have the same amount, because you can give each square a natural number. This is called to numerate.
It is better to look at sets. The sets of the natural and square numbers are equivalent.
There is no doubt that there are more natural numbers than square numbers up to a certain number. This statement is on the view of set theory: The set of the square numbers is a subset of the natural numbers. 

To the second sequence
The sequence of the square numbers is also an arithmetic series of second order. A figure follows to explain this.
In every new row you find the difference of two numbers a row higher. 
The second line is the sequence of the odd numbers. The feature is that you reach a constant difference 2 after two steps. 
Hence the sequence of square numbers has constant differences of differences. 

To the third sequence
You get the second line by forming the following sums:
s4=1²+2²+3²+4²=30, ...
Generally it is  sn=1²+2²+3²+...+n²=n(n+1)(2n+1)/6 .

Here is a proof, which is unusual.
Ansatz: sn=An³+Bn²+Cn s1=1 gives 1=A+B+C
s2=5 gives 5=8A+4B+C
s3=14 gives 14=27A+9B+3C
A=1/3, B=1/2, C=1/6
This gives sn=(1/3)n³+(1/2)n²+(1/6)n = n(n+1)(2n+1)/6.
(4, page 28f.)

To the 4th and 5th sequences
The 4th sequence is 1/1, 1/4, 1/9, 1/16, ... , 1/n². It has the limit 0 for n over all limits.
The associated series 1/1+1/4+1/9+1/16+ ... +1/n²is interesting. 
It is convergent and has the limit pi²/6 (Euler) or rounded 1.65.
(3, page 93f. and page 241)

The series of prime numbers developed by the prime numbers (the sum of 1/p) is divergent. Hence you can imply that there are more prime numbers than square numbers for n over all limits (2, page 322).

Sums    top
Sums of the odd numbers
The sum of the first odd numbers is a square number. There is 1+3+5+...+(2n-1)=n².
Example: 1 + 3 + 5 + 7 + 9=5².

Difference of two square numbers 
Each prime number p>2 can be written as the difference of two square numbers. 
Example: 37=361-324=19²-18²

Division by 8
If you divide the square number of an odd number by 8, you get the remainder 1. 
Example: 19² : 8 =361 : 8=45+1/8

Lagrange's four-square theorem
Each natural number can be written as the sum of four square numbers at least. 
Example: 85=64+16+4+1

Fermat's two-square theorem
A prime number with p=4n+1 can be written as the sum of two square numbers. 
Examples: 5=1²+2², 13=2²+3², 625=7²+24²=15²+20²

Pythagorean triple 
There are triples of three integers, which follow the formula a²+b²=c². You can read more on my German page 3-4-5-Dreieck

Playing with numbers top
I show results of Lietzmann's book (1) from 1948 among other things in this chapter, extended and controlled by my computer. 

Calculation of 51² to 59²
Example: 56²=3136
In words: Add 25 to the ones digit and attach the square of the ones digit. 

Calculation of three-digit numbers with 0 on the tens place.
Example: 203²=041209, 609²=370881
In words: The square numbers of the hundreds digit and the square numbers of the ones digit are external. The double product of the hundreds digit and the ones digit are in the centre. If the double product is three-digit, you must add 1 to the square of the hundred digit. 

Cube numbers are written in the digits 0 to 9. No digit is used twice or more. 
There are 77 numbers in Lietzmann's book. 
1026 753 849=32043² is the smallest number.
9814072356=99066²  is the largest number.
The ten-digit numbers with different digits are called "pandigital numbers".

Cube numbers are written in the digits 1 to 9. No digit is used twice or more. 
Two examples: 139845276=11826² und  923187456=30384²
The nine-digit numbers with different digits are called "zeroless pandigital numbers".

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

Palindromes among the square numbers
698896=836² 1002001=1001²

Five sequences of square numbers 
133²=176 89
1333²=1 776889
13333²=1777 68889

Kaprekar numbers
A number like 703 is called Kaprekar number, because it has the following curious feature. 
If you square the number (703²=494209) and separate the square in 494 and 209, the sum 494+209 is 703 again.
The numbers  1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777 and  9999 are the Kaprekar numbers <10.000. 

Figurate Numbers  top
The square numbers belong to the figurate numbers.
Triangular numbers
Square numbers
Pentagonal numbers
Hexagonal numbers
Heptagonal numbers
Octagonal numbers

1 3 6 10 15 21 28, 36... 
1 4 9 16 25 36 49, 64...
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...
There is the problem in finding the square numbers among them. 

You find at MathWorld: 
Squares among the triangular numbers: 1, 36, 1225, 41616, 1413721, 48024900, ... Sloane's A001110
Squares among the pentagonal number: 1, 9801, 94109401, 903638458801, 8676736387298001, ... (Sloane's A036353)

Square Numbers on the Internet    top


Andreas Göbel 
Ziehen einer Wurzel - Papier Bleistift Methode

Quadratzahlen, Zahlenakrobatik

Tino Hempel 
Das schriftliche Ziehen einer Quadratwurzel

Quadratzahl, Quadratwurzel, Zentrierte Quadratzahl, Wurzel (Mathematik), Schriftliches Wurzelziehen


Eric W. Weisstein (MathWorld)
Square Number, Squarefree, Squareful, Antisquare Number, Square Triangular Number, Pentagonal Square Number
Heptagonal Square NumberOctagonal Square Number, Square Root, Square Root AlgorithmsNewton's Iteration

Patrick De Geest (World of  Numbers)
Extraordinary squares and powers (non-palindromic allowed)

Neil J. A. Sloane 
Integer Sequences
The squares: A000290

Square number, Polygonal_number, Triangular square number, Pandigital_number, 9814072356 (number)
Square root, Methods of computing square roots, Euler's four-square-identityKaprekar number
Lagrange's four-square theorem

References   top
(1) Hermann Schubert: Mathematische Mußestunden, Berlin 1941 (Leitwort: "Ludendo discimus")
(2) Heinrich Behnke u.a.(Hrsg.): Mathematik 1, DAS FISCHER LEXIKON, Frankfurt am Main 1964 
(3) Jean-Paul Delahaye: Pi - die Story, Basel 1999 [ISBN 3-7643-6056-9]
(4) Maximilan Miller: Gelöste und ungelöste mathematische Probleme, Leipzig 1982 
(5) August Schmid u.a.: Lambacher/Schweizer LS9, Stuttgart 1996 [ISBN 3-12-730740-3]
(6) Forsythe, Keenan, Organick, Stenberg: Programmieren mit BASIC, Braunschweig 1976 [ISBN 3-594-10854-6]
(7) Walter Lietzmann: Sonderlinge im Reich der Zahlen, Bonn 1948

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