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Square Numbers


In this article we are going to discuss square numbers.

A couple of lines down you can find our calculator which can square any real number.

The terms square number and perfect square are synonym.

It’s called perfect square because the number is the square of an integer (whole number).
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Square Numbers Definition

A square number is the product of some integer with itself.



Therefore, it is expressed as n2 = n × n; n is a whole number.

As you can see, a square number is an exponentiation involving the base n and the exponent 2, sometimes written as n^2.

In other words, it means n to the power of 2.

It follows from that that n can be positive or negative (If you square a negative number the result will be a positive), but usually the meaning is n > 0.

If n represents the length of a side, then n^2 expresses the area of the shape of a square with side length n.

This answers the question how do you square a number?

Square Numbers Formula

Every square number equals the sum of the first n odd numbers: .

Next, let’s look at some examples.

Square Numbers Examples

  • Example 1 – The square number of 3: 3 × 3 = 9.
  • Example 2 – The square number of 4: 4 × 4 = 16.
  • Example 3 – The square number of 12: 12 × 12 = 144.
If n2 is the square number, then n is the principal square root.
  • Example 4: The square root of 169 (13 × 13) is +/− 13.
To find out if a given integer is a perfect square, calculate its square roots.

If the square root is a whole number (has no decimal places), then the number is indeed a square number.
  • Example 5: Is 18 a square number? No, √18 = ±4.2426406871 (has decimals).
  • Example 6: Is 25 a square number? Yes, √25 = ±5 (has no decimals).
  • Example 7: If the square of a number is y, then what is the original number x? By definition, the product y = x2, so x = √y.
This table is a shortcut to the most searched items in Google: In the section ahead we discuss the frequently asked questions.

Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.

What is a Square Number?

A square number or perfect square is the result when a whole number has been multiplied by itself.

How to Square Numbers in Excel?

To square a number n in Excel type n^2 or n*n.

What is the Square of a Number?

The square of a number is the outcome of multiplying an integer by itself.

How to Do Square Numbers?

You can square a number by means of exponentiation with base n and index 2, or you can multiply n × n.

What Does Squared Mean in Math?

“Squared” is the past tense of the verb “to square”; in math it means that a number has been multiplied by itself.

How to Square Numbers?

Take the decimal, fraction or whole number and multiply it by itself! Note that only if you square an integer do you get a perfect square.

What is a Square Number of 100?

100^2 = 100 x 100 = 10,000.

Is 18 a Perfect Square?

√18 = 4.2426406871 is not an integer, and as such 18 cannot be a perfect square.

What Are the First 10 Square Numbers?

The first 10 square numbers are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81.

Why is 3 Not a Square Number?

A square number can not end in 2, 3, 7 and 8.

What Is the Smallest Square Number?

The smallest square number is equal to 0.

What Is the Easiest Way to Find a Square Number?

The easiest way to find a square number is multiplying a small, whole number by itself.

Why is 20 Not a Square Number?

√20 = 4.4721359549 has digits and as such can not be a square number.
If something remains unclear do not hesitate getting in touch with us.

We are constantly trying to improve our site, and truly appreciate your feedback.

Ahead is the table in the context of this site.

Square Numbers List

The table below contains the square numbers up to 100:
NumberSquare NumberSquare Root
000
11+/− 1
24+/− 1.4142135624
39+/− 1.7320508076
416+/− 2
525+/− 2.2360679775
636+/− 2.4494897428
749+/− 2.6457513111
864+/− 2.8284271247
981+/− 3
10100+/− 3.1622776602
11121+/− 3.3166247904
12144+/− 3.4641016151
13169+/− 3.6055512755
14196+/− 3.7416573868
15225+/− 3.8729833462
16256+/− 4
17289+/− 4.1231056256
18324+/− 4.2426406871
19361+/− 4.3588989435
20400+/− 4.472135955
21441+/− 4.582575695
22484+/− 4.6904157598
23529+/− 4.7958315233
24576+/− 4.8989794856
25625+/− 5
26676+/− 5.0990195136
27729+/− 5.1961524227
28784+/− 5.2915026221
29841+/− 5.3851648071
30900+/− 5.4772255751
31961+/− 5.5677643628
321024+/− 5.6568542495
331089+/− 5.7445626465
341156+/− 5.8309518948
351225+/− 5.9160797831
361296+/− 6
371369+/− 6.0827625303
381444+/− 6.164414003
391521+/− 6.2449979984
401600+/− 6.3245553203
411681+/− 6.4031242374
421764+/− 6.4807406984
431849+/− 6.5574385243
441936+/− 6.6332495807
452025+/− 6.7082039325
462116+/− 6.7823299831
472209+/− 6.8556546004
482304+/− 6.9282032303
492401+/− 7
502500+/− 7.0710678119
512601+/− 7.1414284285
522704+/− 7.2111025509
532809+/− 7.2801098893
542916+/− 7.3484692283
553025+/− 7.4161984871
563136+/− 7.4833147735
573249+/− 7.5498344353
583364+/− 7.6157731059
593481+/− 7.6811457479
603600+/− 7.7459666924
613721+/− 7.8102496759
623844+/− 7.874007874
633969+/− 7.9372539332
644096+/− 8
654225+/− 8.0622577483
664356+/− 8.1240384046
674489+/− 8.1853527719
684624+/− 8.2462112512
694761+/− 8.3066238629
704900+/− 8.3666002653
715041+/− 8.4261497732
725184+/− 8.4852813742
735329+/− 8.5440037453
745476+/− 8.602325267
755625+/− 8.6602540378
765776+/− 8.7177978871
775929+/− 8.7749643874
786084+/− 8.8317608663
796241+/− 8.8881944173
806400+/− 8.94427191
816561+/− 9
826724+/− 9.0553851381
836889+/− 9.1104335791
847056+/− 9.1651513899
857225+/− 9.2195444573
867396+/− 9.2736184955
877569+/− 9.3273790531
887744+/− 9.3808315196
897921+/− 9.4339811321
908100+/− 9.4868329805
918281+/− 9.5393920142
928464+/− 9.5916630466
938649+/− 9.643650761
948836+/− 9.6953597148
959025+/− 9.7467943448
969216+/− 9.7979589711
979409+/− 9.8488578018
989604+/− 9.8994949366
999801+/− 9.9498743711
10010000+/− 10
As follows, the sequence of square numbers is 0 (0 x 0), 1 (1 x 1), 4 (2 x 2), 9 (3 x 3), 16 (4 x 4), 25 (5 x 5), 36 (6 x 6), 49 (7 x 7), 64 (8 x 8), 81 (9 x 9), …

Square Numbers Calculator

Here we inform you that our tool at the top of this page works both ways; you can either fill in the upper or the lower input field.

Our app works with all real numbers, and you don’t need to press a button unless you want to start over.

You may use the up and down arrows (called spinners) to increase or decrease the input value.

Frequently calculated terms include, for example: We now show you two recursive formulas plus an identity which are sometimes useful to come up with n^2.

Additional Information

In addition to the formula discussed above, any product n2 can be produced recursively:
  1. n2 = (n − 1)2 + (n − 1) + n = (n − 1)2 + (2n − 1)
  2. n2 = 2 x (n − 1)2 − (n − 2)2 + 2
This is the difference between two perfect squares:
  • n2 − (n − 1)2 = 2n − 1
Squares of even integers are even:
  • (2n)2 = 4n2
Squares of odd integers are odd:
  • (2n + 1) = 4(n2 + n) + 1
Last digit:
  • In base 10, a square number cannot end in digits 2, 3, 7, 8.
Note that you can always employ the search form located in the menu and in the sidebar of this site to locate information about a particular term.

Or simply consult our square numbers chart.

Next is the summary of our article.

Bottom Line

A square number n2 means n × n, n is an integer and × is the multiplication symbol.

The exponentiation form n2 or n^2 is mostly used to express a square number.

For a number to be a perfect square it’s last digit must be 0, 1, 4, 5, 6 or 9; else it is an imperfect square.

If the second root of a squared number has no decimal places, then the squared number is a square number!

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