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Well, this question is basically a method to point a loophole in Maths which I'm not sure how it is possible...

Let us take a number A and another no. K.
A + A + A + A + A........(k times) = K * A

Everything is all right till here. But the problem arises in the next step.

We know taking A = $\sqrt{3}$ and K = $\sqrt{2}$, the answer is $\sqrt{6}$

But how can we add $\sqrt{3}$ $\sqrt{2}$ times ?

Or, to reframe the question, how can we find the answer of multiplication of two irrational numbers when we know it is not possible to add a number (or infact, do anything), irrational times to itself.

Using the fact that multiplication is repeated addition.

Basically, the above points out that multiplication isn't repeated addition. (Andre Nicolas)

Andrés E. Caicedo
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Nib
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    The post points out that multiplication is not iterated addition. Perfectly true. – André Nicolas Sep 13 '14 at 05:09
  • I know, I was a little surprised when I thought about it. So, I thought it's better to discuss it with more intelligent people @AndréNicolas – Nib Sep 13 '14 at 05:10
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    Previously: If multiplication is not repeated addition, what is it? –  Sep 13 '14 at 05:11
  • If you identify real numbers with equivalence classes of Cauchy sequences (this is a standard construction of $\Bbb R$), multiplication of reals amounts to pointwise multiplication of their representative sequences. One checks this is well-defined, obeys all the original rules set forth for multiplication of integers and then rationals, and indeed is continuous when we view $\Bbb R$ as the metric completion of $\Bbb Q$. – anon Sep 13 '14 at 05:12
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    @Nib: Not more intelligent, but have been around mathematics longer. – André Nicolas Sep 13 '14 at 05:14
  • Why do you have to make your example so complicated? Couldn't you make the same point by asking why $\frac12\cdot\frac13=\frac16$? Or is that somehow less problematic, and if so, why? – bof Sep 13 '14 at 05:20
  • I could, @bof but not all fractions are irrational, and it's possible to add a number fractional times[in most cases] but when you make it the roots of a number, the game changes. Even if you replace $\sqrt{3}$ by any other irrational root and vice-versa , the process would still be wrong. – Nib Sep 13 '14 at 05:24
  • What do you mean? How do you add a number to itself, say, three and a half times? – Dave Sep 13 '14 at 05:25
  • Instead of iterated addition, you can think of multiplication as scaling a diagram. For example, if I multiply $2$ by $3$, we can think of this as scaling the line segment of $2$ unit length by $3$ times. So multiply $\sqrt2$ by $\sqrt3$ is essentially scaling the line of $\sqrt2$ length $\sqrt3$ times. It's sometime like graphical transformation. See Better Explained for an intuitive guide to arithmetic. – Happytreat Sep 13 '14 at 05:27
  • @Happytreat how exactly does one plan to scale something $N$ times, without even knowing the value of $N$(ie. $\sqrt{3}$ is irrational) – Nib Sep 13 '14 at 05:29
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    Well, this question is liken to asking how does one use $\pi$ to calculate the area of say a circle when we don't actually know what exactly it is. In reality, we may not be able to actually measure the square of root of $2$ or $3$ (or for the matter of fact, for any irrational number) but we can do it in Math (abstractly). – Happytreat Sep 13 '14 at 05:32
  • And to re-assure everyone here, the question http://math.stackexchange.com/questions/64488/if-multiplication-is-not-repeated-addition has answers that are pretty good, but none of them consider the argument of irrational numbers... – Nib Sep 13 '14 at 05:32
  • Nib, there are rational numbers that are very close to $\sqrt{3}$ (as close as you like, in fact). You scale in a way that's bigger than all the rational numbers below $\sqrt{3}$, but smaller than all the rational numbers above $\sqrt{3}$. – Dave Sep 13 '14 at 05:33
  • @dave : considering I do do that, then also, you're adding the length of line segments to obtain a bigger line segment of length N. But, think about it, something that is true and possible in geometry, fails in numeric system. Doesn't this suggest something in our Maths is badly flawed? – Nib Sep 13 '14 at 05:36
  • "none of them consider the argument of irrational numbers..." I actually checked for that before I posted the link. See the last few paragraphs of the accepted answer by Arturo Magidin, beginning "Then we move on from the positive rationals (fractions) to the positive reals. This is more complicated, as it involves "filling in gaps" between rationals. ..." –  Sep 13 '14 at 05:40
  • I'm afraid I don't understand your comment. Also, I think you should consider seriously what people are telling you: your remark that multiplication is not just repeated addition applies to any number that isn't an integer. So it applies to 1.7 as much as it does to $\sqrt{3}$. You just can't write down $1.7$ copies of something and add them together. – Dave Sep 13 '14 at 05:40
  • @Nib I believe the question linked contains arguments for irrational numbers, e.g. the answer by Arturo Magidin. You can employ the concept of Cauchy sequences, for instance. – awllower Sep 13 '14 at 05:59
  • @Nib- I think the comment you made about the process failing in maths and succeeding in geometry may be backwards.... As far as the scaling argument is concerned, we could scale In a manner as Dave suggested, but that is not what $\sqrt{3}$ IS. No, it does not have a "whole" representation, as 1, 1/2, ... does, but it is still a real value, and obeys the function $a^x$. – user170141 Sep 13 '14 at 06:07

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