My calculator, has done a very strange thing indeed.

[LjSpike]

Just to mke this simpler, im using a CASIO fx83GT PLUS on its standard settings.

 

Now, I got bored in my maths lesson (Honestly, triganomatory, its so boring and so quick to do when you've written out the sohcahtoa).

Anyway, I decided I felt like doing the square root of 10! Ooooh, the deadly root of 10.

Now I expected either math error, or a long recurring decimal. I then end up with: 3.16227766

Ok, then I decide to do 3.16227766² (squared that is). And I get, 9.999999999 (It didn't show a recurring dot, but its enough decimals for me to know its recurring).

 

So whats going on here?

SQUARE ROOT of 10 = 3.16227766

but 3.16227766 SQUARED is not 10?

 

I then decide trial and error on finding what I would square to hit 10.

Now, if it wasn't interesting enough, that apparently squared isn't the exact inverse of square root? Well, its getting more interesting.

I hit 3.1622776604 as my thing to square to make 10, but then I took it further, I decided to do 3.1622776605 and 3.1622776603 as well, they also make 10.

So after a bit of testing out in the spare time I had because im quick at a lot of geometry, I got the result:

 

The root of 10 is 3.16227766 BUT to result in an answer of 10 when squaring a number the number in question must be

3.162277660x where x is either 1, 2, 3, 4, 5, 6, 7, 8 or 9

 

So how can the square root of something, not result in the original number when squared, yet multiple numbers squared can result in the same answer?

[Klaynos]

Calculators are not perfect. They do rounding, in both the calculation and in presentation of the result.

 

What you're seeing are rounding errors.

[Delta1212]

9.999999999... does equal 10, though.

[Thorham]

9.999999999... does equal 10, though.

 

Yes, but that doesn't exist in normal floating point formats.

[Sensei]

Try this in Windows scientific calculator (you might need enable option in menu, default is basic one):

10 power 0.5 = 3.1622776601683793319988935444327

 

Power ^2 it back, 10!

 

Repeat it:

10 power 0.5 = 3.1622776601683793319988935444327

this time copy to clipboard (ctrl-c),

Open new calculator application.

Paste it, ctrl-v.

Power ^2 = 9.9999999999999999999999999999955

 

In different OS-es it might work differently (different calculators).

But that's what I have in Windows XP 32 bit at least.

Later this day, I will check how it's in Windows 7 64 bit.

Edited October 2, 2015 by Sensei

[LjSpike]

Calculators are not perfect. They do rounding, in both the calculation and in presentation of the result.

 

What you're seeing are rounding errors.

It would explain the fact of it square rooting, then squaring the result of the root, being not perfect, but I would have expected to get it in a "0.34x10 to the power of a very big number" format appear. Plus, if it were that, i wouldn't expect just two more decimal places to make it be able to return to 10.

[ajb]

It would explain the fact of it square rooting, then squaring the result of the root, being not perfect...

As already stated, it is just a rounding error when using a calculator.

 

My calculator which is more powerful than yours gives me that square root 10 to 30 places is

 

3.1622776601683793319988935444327185337195551393252

 

(I can add more places if one wishes)

 

So I now square this and get (including a few more places)

 

9.9999999999999999999999999999999999999999999999998935776

 

You see I do not quite get 10.

 

The reason is that the number that (positive) number that squares to 10 is [math]\sqrt{10} [/math]. Unfortunately, this number is irrational and so cannot be expressed exactly in terms of a decimal expansion.

[LjSpike]

As already stated, it is just a rounding error when using a calculator.

 

My calculator which is more powerful than yours gives me that square root 10 to 30 places is

 

3.1622776601683793319988935444327185337195551393252

 

(I can add more places if one wishes)

 

So I now square this and get (including a few more places)

 

9.9999999999999999999999999999999999999999999999998935776

 

You see I do not quite get 10.

 

The reason is that the number that (positive) number that squares to 10 is [math]\sqrt{10} [/math]. Unfortunately, this number is irrational and so cannot be expressed exactly in terms of a decimal expansion.

But, we have a slight problem here. Ok, I can accept your calculator can divide to more decimal places in an answer, giving a more accurate response, BUT, I could square anything from 3.1622776601 to 3.1622776609 to equal to 10. But if a more accurate version of 10 squared is

3.1622776601​68 then surely only

3.1622776601​ to 3.1622776602 should result in an answer of 10, OR they should both not equal 10, as those two rational numbers aren't the square root of 10. 3.1622776601​ should result in just under 10 and 3.1622776602 should be just over 10, and then 3.1622776609 should be quite a way away from 10, yet ALL of these numbers are resulting in 10 when squared?

 

 

[John Cuthber]

Imagine my calculator only has 2 places of decimals and I try to find the square root of ten.

It's going to tell me that it is 3.16.

But 3.16 squared isn't ten, it is 9.8956

So, if I put in 10, then press the square root key, then square the result I should get 9.9856, but the calculator can't display that so it will either say 9.99 or 9.98 depending on whether it truncates or rounds off.

 

How could it give the right answer?

It's certainly not a very strange thing for a calculator to do.

[overtone]

http://www.stewartcalculus.com/data/default/upfiles/LiesCalcAndCompTold.pdf

 

Rule of thumb: when learning calculus, set your calculator aside until all the algebra has been done. Do not take roots to decimals, do not take fractions to decimals, do not take trigs to decimals, and most definitely do not subtract these kinds of decimals from each other, until all algebraic manipulations possible have been accomplished.

 

Calculus is analog, your calculator is digital. Be warned.

[ajb]

...yet ALL of these numbers are resulting in 10 when squared?

It is still something to do with rounding off and how your calculator has been programmed.

 

3.1622776601​ should result in just under 10....

 

9.999999999567532

 

...3.1622776602 should be just over 10

10.000000000199988

 

 

...and then 3.1622776609

10.000000004627175

 

 

So clearly all answers are close to 10 and it is not surprising a basic calculator gives the value 10.

 

Again, it is something to do with how your calculator works and nothing deep.

Edited October 4, 2015 by ajb