Why is the square root of a decimal less than one greater than the number that is being squared?

Question

I was wondering about this when I was being taught maths the other day. I got the answer 'it just is' which hardly seems helpful. Someone else tried (and confused) me by making a graph. So I'd like a simple answer please.

A simple example of what I mean is:

The square root of 0.16 = 0.4.

0.4 is greater than 0.16

I was wondering why this doesn't fit the pattern of the square root being less as is the case with integers greater than 0.

Thank you for your answer(s) in advance. =]

Answer 1

A decimal can be interpreted as (1/Z) where Z is any random integer.

Let's just say for 0.1111...., it is equal to (1/9).

Since decimal exists as 1/Z,  everything becomes inversed when you try to perform regular functions on it. When you perform a root function to this 1/Z, you are rooting the denominator, which makes it overall greater (Remember 1 rooted or squared is the same thing)

However, a square root can actually be less, if you factor in the negative root. Rooting (1/9) for instance actually yields positive and negative (1/3).

Answer 2

One way by which you can visualize this easier is to look at the square root of a fraction less than 1 say 1/64:

sqrt(1/64) = sqrt(1)/sqrt(64) = 1/8

1/8 is definitely greater than 1/64

You note that the divisor of 1 becomes less when the square root is extracted. It follows therefore that the square root of a positive number less than 1 is greater than the number.

Answer 3

Think of it this way - when you find the square of a number less than one, what happens to the result? It decreases, unlike numbers > 1, which always increase. The square root is the inverse of squaring a number. So, if we square a number less than 1 and the answer goes down, because the square root is the inverse of squaring a number, surely the answer should go up?

The same principles apply with multiplication and division. Normally, multiplying a number will make it increase - unless it is smaller than 1, then it will go down. Division is the inverse of multiplication, and therefore if you divide by a number less than 1 the answer will go up - exactly the opposite to what would happen in a multiplication.

Hope this helps.

Answer 4

The square root of a number (lets call this known number A) is another number (lets call this unknown number B) that, when multiplied by itself, equals the known number A.

For example:

If A = 100 (with no sign indicated, it is understood to be positive), B = +/- 10.  This means that, if you multiply +10 times +10, the answer = +100.  Also, if you multiply -10 times - 10, the answer is also +100.

To simplify this answer, I will leave the negative number out of the narrative!  Just remember that the square root of a positive number has two roots, one positive and one negative.

Examples:

The square root (sqrt) of 10,000 equals 100 (since 100 times 100 = 10,000)  

Sqrt of 100 = 10 (10 times 10 = 100)

Sqrt of 49 = 7 (7 times 7 = 49)  

sqrt of 4 = 2 (2 times 2 = 4)

sqrt of 1 = 1 (1 times 1 = 1)

sqrt of 0.81 = 0.9 (0.9 times 0.9 = 0.81)

sqrt of 0.16 (your example) = 0.16 (0.4 times 0.4 = 0.16)

sqrt of 0.01 = 0.1

sqrt of 0.000001 = 0.001

The square root of ANY number A less than one (1.00) equals a number B that is greater than number A because, when the decimal number B is multiplied by itself, the decimal placement required will automatically assure that number A is less than B ( or B is greater than A).

Answer 5

think of it as a fraction where numerator is less than denominator ...

when we square the fraction, we square both numerator & denominator

but numerator is smaller than denominator

leaving a smaller fraction (even more range between top & bottom)

Answer 6

the only way I can think to explain is

looking at it in reverse (.4^2)

Remember your multiplying two decimals so the  answer will have the total number of  decimal places

just like .2 * .2 not .4  but .04

so, the oppsite is true with roots. Help any? I hope a little.

Answer 7

Any number multiplied by a number less one, is less than the number being multiplied. 10 multiplied by 0.5 is 5, which is less than 10. Any number multiplied by more than one, will be more than the number being multiplied.

A number that is more than one, multiplied by itself, will allways be more the orginal number, and therefore also more than one. Therefore the square root of a number that's less than 1, must also be less than 1.

Herein comes the first rule, namely any number multiplied by another number which is less than one will be less than the orignal number. That's why 0.4 multiplied by 0.4, will be less than 0.4.

Hope this helps.

Maybe the confusion comes in, because in simple everyday life, we typically chose our decision whether to multiply or divide, on whether the second number is greater or smaller than one.  Any multiplication between two numbers can be written as division. For example, we rather say 100 divided by 2, than 100 multiplied by a half, even though they are equivalent. The converse is also true, we'd rather say 100 times 2, than 100 divided by a half (which is also 200). Because we allways try to "keep" the second number above one, we falsely think of multiplication to allways make things more, and division to make things less.

Answer 8

i'll give it a try!

well say if you had 0.5 squared that would be the same as 0.5 x 0.5 and that is only half of the half which is a quarter so..

0.5 x 0.5 = 0.25

the answer was less than it started and this is the case with all decimals.

now if you want the square root of a decimal number it is exactly the same but in reverse but the same reason still aplys

0.4 x 0.4 =0.16

because it is only 2/5 of 2/5 which equals a lesser number

i hope that helps! if you need any further explaining just ask!

Answer 9

i guess it is just because of the decimal..

like ur given eg.

square root of 0.16 = 0.4.

square rooting 16/100 = 4/10

see the fraction expressed.. the value of denominator is larger than numerator.. when u square root it, (square root16) / (square root100)

thus making the denominator smaller at the same time..

eg. numerator: square root16 = 4

denominator: square root100 = 10

however, on the other hand, when square rooting numbers greater than 0, the denominator is always smaller than numerator...

take 6.25 for example..

fraction = 25/4

square root 25/4 = 5/2

u can see the pattern now???

square rooting larger values of numbers always gives smaller value..

and square rooting smaller values of numbers eg. 2,3 doesn't give as small range of values compared to square rooting 100 or 10000!

try looking at it this way!

Answer 10

Let k > 1.  Then we obviously have that:

k² > k

Rearranging both sides:

1/k > 1/k²

(1/k) > (1/k)²

If we have a number m < 1 we can write it as m = 1/k for k > 1.  Therefore, we have that:

(1/k) > (1/k)²

m > m²

Taking the square root of both sides:

√m > m

Hope this helps!