Making sure we know what a square of a number is first. a2 represents the number "a" squared ...
that is to say a*a ... or a times a. This could be
22 ==> 2 squared which is 2 times 2 which is 4
32 ==> 3 squared which is 3 times 3 which is 9
42 ==> 4 squared which is 4 times 4 which is 16
52 ==> 5 squared which is 5 times 5 which is 25
That group of numbers ... 4,9,16,25, etc etc are called "perfect squares".
There are a "slew" of perfect squares. In fact any whole number multiplied by itself produces a "perfect square".
So,
502 ==> 50 squared which is 50 times 50 which is 2500 and 2500 is therefore a "perfect square".
Here is a list of numbers, and that number squared (which is a "perfect square").
Perfect
Number Square
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625
Now to talk a minute about "inverse operations". This means "opposite operations".
Plus and minus are inverse operations. Pick a number (as long as it is 35) ... then add 5 and then subtract 5.
What do you now have? Well ... 35 of course. We added 5 and subtracted 5 which just "neutralized" each other.
Adding and subtracting are "inverse operations".
Likewise multiplication and division are "inverse operations". Take a number (as long as it is 8).
Multiply it by 4 and divide by 4. What do you have? Well ... 8 of course.
We multiply 8 by 4 and get 32 then divide 32 by 4 and we get 8. Inverse operations.
There is another set of "inverse operations" and this is squaring a number and then taking the square root.
Pick a number (as long as it is 9). Then square it and take the square root of the result.
OK ... I pick 9 ... square it and I have 81 ... then take the square root of 81 and we have 9.
So, squaring a number and taking the square root of it are inverse (opposite) operations.
And it does not matter which order this is in. Pick a number (as long as it is 16).
Take the square root and that would be 4. Square 4 and that would be 16.
So, back to this concept of perfect squares and inverse operations.
Pick a number (as long as it is 15).
Square 15 and you have 225, which is referred to as a "perfect square".
Take the square root of this "perfect square" (225) and what do you have?
Well, 15 of course. So, if you take a square root of a perfect square you return to the number from
which the "perfect square" came. That is to say, by example:
Perfect
Number Square Number SqRt
2 4 4 2
3 9 9 3
4 16 16 4
5 25 25 5
So, now you should understand what a perfect square is and that squaring a number and taking the square root of the
resulting number are inverse operations. If not, pick a number and I don't care what it is. And I mean a "whole number"
(lets make it easy anyway). Square that number and record the value. That "recorded value" is a perfect square.
Take the square root of this perfect square and you will return to your original number.
To explain, I pick 40. Squaring 40 I get 1600 and 1600 is therefore a "perfect square".
If I take the square root of 1600, I get back to my original number, which was 40.
Squaring a number and taking the square root of the resulting number are
inverse operations.
OK, now let me reproduce a portion of that table from above.
Perfect
Number Square
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
As is obvious, taking the √(PerfectSquare) will give us a whole number. By that I mean
√36 = 6 AND
√49 = 7 AND
√64 = 8 AND
Note: the symbol √ is the square root sign. It is not a "full" square root sign,
and could likely be better if it was "more full" in appearance, but for the purposes of this
article, it is what we have.
But what about √40 ?
Well, just from observation from above it MUST be between 6 and 7 since 40 is between 36 and 49, right??
And actually that is right, but it WILL NOT be a whole number.
In fact the √40 = 6.32 (to the nearest hundredths) .
How bout the √50?
Well, 50 is ever so slightly larger than 49,
so the square root must be "ever so slightly" larger than 7.
And in fact it is. √50 = 7.07, rounded to two decimal places.
And guess what? √63 must be ever so slightly SMALLER than 8. And it is
√63 = 7.94, rounded to two decimal places.
In case you need to view this from a different angle then try it from the square root
back to the value of the square root.
For example, 4 is a perfect square and its square root is 2
and also 9 is a perfect square and its square root is 3.
And all the numbers between 4 and 9 have square roots between 2 and 3.
Also note you can use the square roots that you easily know (the perfect squares)
to predict with some degree of accuracy those which you do not know exactly (all the imperfect squares).
Note: perfect squares are flagged with "**", 4, 16, 25, etc.
SqRoot to nearest
thousandths
--------
√2 = 1.414
√3 = 1.732
** √4 = 2
√5 = 2.236
√6 = 2.449
√7 = 2.646
√8 = 2.828
** √9 = 3
√10 = 3.162
√11 = 3.317
√12 = 3.464
√13 = 3.606
√14 = 3.742
√15 = 3.873
** √16 = 4
√17 = 4.123
√18 = 4.243
√19 = 4.359
√20 = 4.472
√21 = 4.583
√22 = 4.690
√23 = 4.796
√24 = 4.899
** √25 = 5
√26 = 5.099
√27 = 5.196
√28 = 5.292
√29 = 5.385
√30 = 5.477
√31 = 5.568
√32 = 5.657
√33 = 5.745
√34 = 5.831
√35 = 5.916
** √36 = 6
√37 = 6.083
√38 = 6.164
√39 = 6.245
√40 = 6.325
√41 = 6.403
√42 = 6.481
√43 = 6.557
√44 = 6.633
√45 = 6.708
√46 = 6.782
√47 = 6.856
√48 = 6.928
** √49 = 7
√50 = 7.071
√51 = 7.141
√52 = 7.211
√53 = 7.280
√54 = 7.348
√55 = 7.416
√56 = 7.483
√57 = 7.550
√58 = 7.616
√59 = 7.681
√60 = 7.746
√61 = 7.810
√62 = 7.874
√63 = 7.937
** √64 = 8
√65 = 8.062
√66 = 8.124
√67 = 8.185
√68 = 8.246
√69 = 8.307
√70 = 8.367
√71 = 8.426
√72 = 8.485
√73 = 8.544
√74 = 8.602
√75 = 8.660
√76 = 8.718
√77 = 8.775
√78 = 8.832
√79 = 8.888
√80 = 8.944
** √81 = 9
√82 = 9.055
√83 = 9.110
√84 = 9.165
√85 = 9.220
√86 = 9.274
√87 = 9.327
√88 = 9.381
√89 = 9.434
√90 = 9.487
√91 = 9.539
√92 = 9.592
√93 = 9.644
√94 = 9.695
√95 = 9.747
√96 = 9.798
√97 = 9.849
√98 = 9.899
√99 = 9.950
** √100 = 10
This is an ideal time and ideal subject to distinguish between rational and irrational numbers.
Very simply, rational numbers are numbers which can be expressed as a/b where both a and b
are integers and b is not equal to zero. Any rational number which is expressed as a decimal
will either terminate (such as 1/4 = 0.25) or repeat (such as 1/3 = 0.333 repeats).
Wikipedia has an excellent explanation of rational numbers.
All real numbers which are not rational are, by definition, irrational.
A commonly used example of an irrational number is pi.
Pi relates the circumference of a circle to its diameter and the first few digits are 3.1416,
but it does not repeat and cannot be expressed as a fraction (a/b).
Another frequently used example of irrational numbers are √nonPerfectSquare .
The square root of any non-perfect square is irrational.
Lets look at the √4 and the √5 .
The number 4 is a perfect square as it is the square of 2. And the √4 is equal to EXACTLY 2.
Not approximately 2, but PRECISELY and EXACLTLY 2.
Also, 2 can be expressed as a fraction which would be 2/1 .
Now, by contrast, √5 is not rational, and cannot be expressed as a fraction (a/b) .
And when expressed as a decimal fractional equivalent does not terminate and does not repeat.
Another way of saying all of that is that you never know EXACTLY what it is, akin to pi in that regard.
From the above table √5 = 2.236 but that is only
expressed to 3 decimal places. The following table is √5 expressed to different decimal places:
2.2
2.24
2.236
2.2361
2.23607
2.236068
2.2360680
2.23606798
2.236067977
2.2360679775
2.23606797750
2.236067977500
2.2360679774998
2.23606797749979
2.236067977499790
2.2360679774997897
If you only wanted the sq root of 5 to one decimal place (tenths) then the best answer is 2.2 .
Turning this around, if you square 2.2 then you get 4.84, which is clearly NOT 5.
But if you square the next bigger "tenths" (2.3) then you get 5.29 .
Clearly also NOT 5 and also 4.84 is closer to 5 than is 5.29, so to one decimal place (tenths)
then 2.2 is best guess. In fact here is a table adding one decimal place at a time trying to
figure out EXACTLY what the square root of 5 is.
2.2 4.84
2.24 5.0176
2.236 4.999696
2.2361 5.00014320
2.23607 5.0000090448
2.236068 5.000000100623
2.2360680 5.00000010062390
2.23606798 5.0000000111812804
2.236067977 4.9999999977648717
2.2360679775 5.0000000000009399
2.23606797750 5.0000000000009399
2.236067977500 5.0000000000009399
2.2360679774998 5.0000000000000462
You will note that not a single entry produces EXACTLY 5. This is the nature of irrational numbers.
A couple of things that are rather common when studying square roots is to be required to plot them on a number line.
To refresh our memory, this is a number line.
|-----|-----|-----|-----|-----|-----|-----|-----|-----|
0 1 2 3 4 5 6 7 8 9
Of course the number line goes on forever, but we have to stop
somewhere. And when we are using the number line and also
studying square roots let us add a new feature to it. Just
go ahead and put the sqRoot of the perfect square it goes
with. By that I mean 4 is √16 so just go
ahead and label it as shown below.
|-----|-----|-----|-----|-----|-----|-----|-----|-----|
0 1 2 3 4 5 6 7 8 9
√1 √4 √9 √16 √25 √36 √49 √64 √81
And now, if you are asked to plot 3.5 on the number line,
it is obvious where it goes. And by the same token if
you are asked to plot √12 it is obvious
where it goes. It goes at about the same place 3.5
goes but this time focus on the √PerfectSquares
and it goes between √9 and √16,
since 12 is between 9 and 16. AND, it likely is just about
in the middle of the space between 9 and 16, since 12 is
near the middle of the range from 9 to 16.
If you have a math quiz in which you are likely to have
plot √(PerfectSquares) just go ahead the very
first thing and make you a number line. THEN add the
√(PerfectSquares) at the appropriate places
like above. Now, it is quite possible that you will
be required to learn more than perfect squares up
to 9. I stop at 9 because that is all that will fit.
Go ahead and make a number line up to say, 25.
Of course the numbers which would get matched up
are:
1 √1
2 √4
3 √9
4 √16
5 √25
6 √36
7 √49
8 √64
9 √81
10 √100
11 √121
12 √144
13 √169
14 √196
15 √225
16 √256
17 √289
18 √324
19 √361
20 √400
21 √441
22 √484
23 √529
24 √576
25 √625