# Definition for multiplication [duplicate]

I am serious.

What is the definition for multiplication ?.

Every book and resources online(the one's I have seen) tells multiplication is repeated addition. But that's only some defintion that can be called valid when one of the numbers is a positive integer. For eg: $5 \times 4=4+4+4+4+4=20$ but how can we define $1.4\times 1.5$ or $\sqrt{2} \times \sqrt{2+\sqrt{3}}$ using only the definition?.

I would like to see a better definition

• First deal with natural numbers, then define multiplication by negative numbers, then rationals and then reals... Apr 4, 2013 at 16:22
• Why the "I am serious" at the beginning?
– Pedro
Apr 4, 2013 at 16:31
• @Peter Tamaroff. I just didn't want the question to be neglected by Stack Exchange users thinking this is a joke or something like that. Apr 4, 2013 at 16:36

If you know how to multiply numbers in $$\Bbb Z$$, you know how to multiply numbers in $$\Bbb Q$$. Let $$x,y\in\Bbb R$$. We may define $$x\cdot y=\lim x_n y_n$$ where $$y_n,x_n$$ are sequences of rationals such that $$y_n\to y$$ and $$x_n\to x$$. It is a nice exercise to prove that the resulting number $$x\cdot y$$ is independent of the choice of $$y_n,x_n$$. We may define the sum analogously, as $$x+y=\lim(x_n+y_n)$$.

The above is rather more intuitive. You can also define a real number as proper nonempty susbset of $$\Bbb Q$$ that has no maximum and is unbounded below. The above might be put as

D A real number $$\alpha$$ is a subset of the rational numbers $$\alpha$$ such that

$$(1)$$ $$\alpha\neq \Bbb Q,\alpha\neq \varnothing$$

$$(2)$$ If $$r\in\alpha$$ there is a $$p\in\alpha$$ for which $$r.

$$(3)$$ If $$r\in \alpha$$ and $$p then $$p\in \alpha$$.

One can then define addition and multiplication of these cuts and construct the field of real numbers $$\Bbb R$$.

The definition you have quoted holds true for positive integers.

If it's the reals you have in mind, there's no way around a proper construction of the real numbers, which includes defining the operations.

For the natural numbers indeed we define multiplication as repeated addition. But as you note, this is not really the case when you change the settings and allow negative numbers, fractions and all sort of real (or complex) numbers.

Indeed one can find Keith Devlin's columns writing why multiplication should not be taught as repeated addition, but as a second primitive operation. This to prevent this sort of confusion, when encountering fractions and roots later on.

But even if multiplication is not repeated addition, this definition for the natural numbers is a lighthouse in the cold and dark space that is undefined mathematics. While wading through the possible reasonable definitions, it is good when one has some lifeline to help them focus the idea of what is a good definition for multiplication.

What I meant by that last paragraph is that we define the multiplication in the complex numbers (and by extension the real numbers, rational numbers, and finally the integers) by drawing from the "repeated addition" definition for the natural numbers.

After defining the natural numbers we add their additive inverses, which give us the integers. We define the multiplication of two integers as an extension, which will respect the laws of addition. So it means that we define $a\cdot b$ in the integers, as $|a|\cdot|b|$ in the natural numbers (repeated addition again), but we also throw in $\operatorname{sgn}(a)\cdot\operatorname{sgn}(b)$ in which we agree in advance the if both the signs are the same then we have $1$, otherwise we have $-1$. Of course if one of $a$ and $b$ are zero then everything is zero and we don't worry about signs. This definition extends the definition of repeated addition, and it is compatible with the new elements of our universe and how they behave with respect to addition.

Next we define the rational numbers, which are easy, by adding multiplicative inverses. That is, we add for every integer (except zero) an object which when multiplied by that integer returns $1$, then we close the entire shindig under multiplication and addition in a way which is compatible with this definition, and lo and behold, we extended by another step again.

Now we want to define the real numbers, and their multiplication and addition. Let us use the definition of the real numbers as equivalence classes of Cauchy sequences of rational numbers. That is, a real number is the limit of a sequence of rational numbers. We define addition, and multiplication, of real numbers as the limit of the pointwise addition (or multiplication) of the sequences converging to them, that is $$x\#y=\lim_{n\to\infty}x_n\#\lim_{n\to\infty}y_n=\lim_{n\to\infty}x_n\#y_n$$ Where $\#$ is one of the operations. Of course we have to check that this is well-defined, but it works out just fine. And so we extend the definitions again. And now we can finally calculate $\sqrt{6+\sqrt2}\cdot\sqrt\pi$.

Lastly we went this far, we can take one final step and define the complex numbers, but there things are easier. Every complex number $z$ is $a+bi$ where $a,b$ are real numbers and $i$ represents $\sqrt{-1}$. So we define addition and multiplication to extend the operations from the real numbers, and we have $$(a+bi)+(c+di)=(a+c)+(b+d)i\text{ and }(a+bi)\cdot(c+di)=(ac-bd)+(ad+bc)i.$$ We have to verify that these extend the definitions from before, but that's even easier than before.

So now we have multiplication defined for the complex numbers, from which we can draw the definition for real numbers, rational numbers, integers, and return to the natural numbers. And because we built everything on top of the natural numbers, there is some sort of coherence. And if we want to fully compute $\sqrt{6+\pi}\cdot\ln\frac12$ then we can take rational sequences converging to each number, then calculate their pointwise multiplication by returning to the integers and then to the natural numbers if we need to.

Of course there are other approaches to mathematics. One can argue from a formalist point of view, that we define the real numbers as a structure with two binary operations which obey certain axioms (and maybe a handful of basic functions like $\sin,\cos$ and so on), and we require that certain properties hold (which make this structure unique).

From these operations and properties we can define, for example $\sqrt{6+\sqrt2}$ and $\pi$, and we don't really compute their product, we just say that it is their product, because we know this product exists and it is unique. Then we may or may not wish to prove that this expression is equal (in this structure) to another expression, e.g. $\sqrt{6+\sqrt2}\cdot\pi=\sqrt{(6+\sqrt2)\cdot\pi^2}$, and so on.

• @boywholived: Oh, it's fine. I capped out today anyway so all the votes on this (or other posts for the next seven hours) are not counted for anyway. I'm glad I could help. Apr 4, 2013 at 17:17