I know that the square of a number is that number multiplied by itself. I know that the square root is the number when multiplied by itself will give you that number.
I don’t understand why the square root of 2 isn’t 1.
But 1x1=1. The square root of 2 is 1.4142135623 to 10 decimal places. Therefore 1.4142135623 x 1.4142135623 should be 2, but it is 1.99999999979325598129
If the square root of 2 is given to 31 decimal places , the square is 1.999999999999999, which is not 2.
Why is 1.4, which when multiplied by itself is 2.8 the square root of 2? Why is 1.414 x 1.414 equal to 1.999999?
I’ll leave a more complete explanation to stouter hearts than myself, but just to start things off 1.4 multiplied by itself is 1.96, not 2.8
To put it in simeple terms, you cannot write the square root of 2 as a decimal. The actual value has a neverending list of numbers after the decimal point. For ease of use, the square root is shortened when doing simple calculations.
Regarding to 31 decimal places, 1.99999999999 may not be 2, but 1.9999999999 recurring (i.e. the list of 9s never ends) can be shown to be more or less 2, but usign bits of Analysis that I never really understood when I did it at University 10 years ago, never mind now.
The square root of 2 is only approximately 1.4142.
1.4142 x 1.4142 = 1.99996164, which is approximately 2.
Where do you get the other stuff about the square root of 2 being 1? Or 1.4 multiplied by itself is 2.8?
First of all, 1.4x1.4 is 1.96, not 2.8.
Second of all, the square root of 2 isn’t 1 because 1 times 1 isn’t 2. That’s the very definition of square root.
Third of all, 1.414x1.414 equals 1.999396, because that’s what you get when you multiply 1.414 with itself.
And fourth of all, I’m not sure I understand your question. Surely you understand that some numbers cannot be exactly expressed with decimals, such as pi or 1/3. I’m not sure the square root of 2 is such a number, but in any case it has a lot of decimals, as you show. Since you haven’t exactly expressed the square root of 2, you have only an approximation, and the square of that approximation obviously won’t be 2.
If I’ve missed the boat entirely, please clarify. I feel like I’m missing something.
The poor guy probably hit the + instead of the x to get 1.4 to 2.8
Give him a bit of break.
Dreaming, dreaming.
It can catch people by surprise that the product of a multiplication of a number by itself (the square) is less than the product of an addition of a number to itself, for numbers less than 2; it’s because the addition formula is linear and the square isn’t.
Of course the simpler explanation is that by adding a number to itself, you’re doubling it, by multiplying it by less than two you are less than doubling it.
An excellent explanation Mangetout - it’s the same one that, for some reason, made my fourth grade math teacher think I was some sort of genius when I explained it to her.
I went on to fail Calculus three times.
Priceguy, you got it, SQRT(2) not only can’t be expressed as a finite decimal, but unlike (1/3) it is also an “irrational” number, one that cannot be expressed as a ratio of two integers, A/B. I believe Pythagoras worked out a proof for that.
In theoretical math classes I would generally NOT multiply it out when getting the answer to a calculation, but leave it expressed as Sqrt(2) (using the sqrt sign, which I don’t know how to encode here). As in, for instance:
cosine 45[sup]o[/sup] = [Sqrt(2)]/2
In practical engineering, you just take it out to howevermany significant digits will do so that the thing you are designing will work and fit within the specified tolerances, and once you’re satisfied it’ll work right you move on.
This is one simple proof:
1.9999…*10 = 19.9999… =>
9*1.9999… = 19.9999… - 1.9999… = 18
=> 1.9999… = 18/9 = 2 =>
1.9999… = 2
QED
There is nothing to add to the above except to confirm that sqrt(2) is not a repeating decimal (like, say, 1/7 = .142857142857…), but just goes on forever with no evident pattern. This was discovered by the Pythagoreans at least 2500 years ago (and their proof is still used today) and it sort of destroyed their philosophy that everything could be done with whole numbers and fractions thereof. Interestingly, sqrt(2), like all quadratic roots, can be expressed as a periodic continued fraction, but I will not attempt to illustrate a continued fraction in this medium. Once I did, but the formatter completely destroyed it, taking out all the carriage returns and spaces I had so carefully entered. In words, the square root of 2 is 1 plus 1 over (2 plus 1 over (1 plus 2 over (1 plus 2 over 1 (plus…)))…) where the 1s and 2s, not to mention the parentheses, go on forever.
I know that the square of a number is that number multiplied by itself. I know that the square root is the number when multiplied by itself will give you that number.
I don’t understand why the square root of 2 isn’t 1.
Let’s start with what you know and do a little thinking here. From your first two statements and basic arithmetic, you know that 1x1=1. That means that the square of 1 is one, and the square root of one is 1. You also know that 2x2=4, which means that the square of 2 is four and the square root of four is 2.
Now, two is somewhere in between one and four, so it’s only logical that with the square root of one being 1, and the square root of four being 2, then the square root of two must be somewhere between 1 and 2. Given that, and the fact that we already know the square root of one is 1, it makes no sense to think that the square root of two might be 1. Surely if you ponder this for a moment you will understand why the square root of two isn’t 1.
*Why is 1.4, which when multiplied by itself is 2.8 the square root of 2? *
The obvious error here is that 1.4 added to (not multiplied by) itself is 2.8. This and your thinking that the square root of two should be 1 suggests that you confuse addition and multiplication. You seem to understand how to multiply, and I assume you understand how to add. It looks like you need to put some effort into thinking about when to multiply and when to add.
As for the rest of the OP, surely you understand the concept of approximation. The square root of two to 7 decimal places is 1.4142136, which multiplied by itself is 2.00000010642496. Whenever you’re dealing with an approximation, you’re going to be a bit over or a bit under the exact value. Did you not know that?
For some reason, a lot of people don’t like this proof. I’ve seen a couple threads here go to four pages while people debate whether .9… is really equal to 1.
Let`s go for FIVE pages here!!!
Yes, ultrafilter, there have apparently been huge threads about infinities and limits that seem to rage on without end. The problem is roughly analagous to:
limit, as k goes to infinity
1 +
k
-----/ 9
\ —
/ 10i
-----
i=0
= 2
(pardon, not good enough at clever board tags to do this other than asciiart).
Or,
limit as k approaches infinity of
2 - 1/k
Drives people nuts. Highly counterintuitive, but necessary for various types of mathematics not the least of which is calculus. However, this is a hijack of the original post…
Drat. That first asciiart one should have been
1+ sum(i goes from 1 to k) of 9/10i
Oh well.
Don’t trust you calculator!
Give him a bit of break?
I think you ment “Give him a bit of a break”.
You can do this: √2 (that’s √)
Unfortunately, such tricks won’t work reliably here because the SDMB have no explicit character encoding (there should be one). So writing sqrt(2) is better.
I have no proof, but I’ve found that many people will find this kind of explanation satisfactory:
1 = 1/3 + 1/3 + 1/3 = 0.333… + 0.333… + 0.333… = 0.999…