What Does Multiplication Really Mean?

[anantchowdhary]

How do we define multiplication

i mean..like if we have sqrt3*sqrt2

We can't say it is adding sqrt3 sqrt2 times...

so wat is multiplication exactly

thanks

 

[DavidWhitbeck]

I think that it's best understood in the context of the definition of a field. It's a binary operator on F such that (1) there exists an identity 1 such that 1*a = a for all a in F (and the multiplicative identity is not equal to the additive identity 0), (2) has closure that is if a,b in F, then ab in F, and (3) associativity (ab)c = a(bc) for all a,b,c in F. Have I missed anything?

You usually use a map from \phi: N to R defined by \phi(1) = 1 and \phi(n+1) = \phi(n) + 1 to flesh it out.

 

[anantchowdhary]

er...thnks..for the reply..but is there any way to explain it in a very basic way..i mean.im at high school

 

[Diffy]

Hopefully you have taken Geometry for this explanation.

Draw an isosceles right triangle with legs of length 1.
The hypotenuse is now sqrt(2)

You now have a line segment of length sqrt(2)

If you draw a line segment of length 1 perpendicular to the endpoint of that line segment, you can form a right triangle with a side of length 1 and another of length sqrt (2). The hypotenuse of that triangle is sqrt (3).

You have now constructed two linesegments, one of length sqrt(2) and another of length sqrt (3). If you takes those line segments you can make a rectangle of base sqrt(2) and height of sqrt(3). Of course you know that the area of a rectangle is base times height. The area of the rectangle you constructed is a representation of what multiplication actually is.

 

[DavidWhitbeck]

er...thnks..for the reply..but is there any way to explain it in a very basic way..i mean.im at high school

Oh... wait, why are you worried about it if you're in high school? Alright I'll try something else.

Alright integers-- you multiply n by m by adding n m times or vica versa.

Alright rational numbers-- if p and q are rational numbers there are smallest integers n,m, r and s such that p=n/m and q=r/s. Define pq by nr/ms, where you compute nr and ms by the method outlined above.

Alright algebraic irrational numbers-- you can define these types of numbers as the smallest real number that's bigger than some set of rational numbers. And you can define multiplication using that.

Let me just do it by example $$\sqrt{2}$$ is the smallest real number such that it's bigger than all rational numbers p such that $$p^2 < 2$$. Similarly you can define $$\sqrt{3}$$ as the smallest real number such that it's bigger than all rational numbers q such that $$q^2 < 3$$. If you take any p that satisfies the first inequality and multiply it by any q that satisfies the second inequality you will have a new number $$r=pq < 6$$. You now have a way to define the multiplication of those two irrational numbers-- it's the smallest real number that's still bigger than all of those r's.

 

[anantchowdhary]

Oh... wait, why are you worried about it if you're in high school? Alright I'll try something else.

Well..i just wantd to know so i thought i would ask


hmm...i don't quiteunderstand...if we multiply using your method..then arent we using conventional rules...i mean..the same as for integer multiplication...then how do we know this applies for irrational numbers also..I know it does but is there any way to PROVE it

thnx again

 

[HallsofIvy]

1. Define multiplication by integers- that's the part you accept, right?

2. Define multiplication by numbers of the form 1/n, n a non-zero integer. m*1/n is the number x, such that nx= m. It's not to difficult to show that such a thing exists and is unique (i.e. that "multiplication by 1/n is well-defined).

3. Define multiplication by numbers of the form m/n, m an integer, n a non-zero integer, that is, all rational numbers, by x*(m/n)= (x*m)*(1/n).

4. Finally, define multiplication by irrational numbers in the following way: If x is irrational number, then there exist a sequence {a_n} or rational numbers that converge to x. xb= lim (a_nb). Again, it is possible to show that sequence always converges.

That last is the one that will give you $\sqrt{3}*\sqrt{2}$. There exist a sequence of rational numbers {a_n} that converges to $\sqrt{3}$ and a sequence of rational numbers {b_n} that converges to $\sqrt{2}$. $\sqrt{3}*\sqrt{2}$ is defined as lim (a_nb_n).

The fact that $\sqrt{3}$ can be represented by the infinite decimal number 1.73205... means that {1, 1.7, 1.73, 1.732, 1.7320, 1.73205, ...} is an infinite sequence of rational numbers (each is a terminating decimal) which converges to $\sqrt{3}$. Similarly, $\sqrt{2}$ can be represented by the infinite decimal number 1.414213... so that {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ...} is a sequence of rational numbers converging to $\sqrt{2}$. That gives a proof that we can multiply irrational numbers "digit by digit".

 

[LukeD]

Well..i just wantd to know so i thought i would askhmm...i don't quiteunderstand...if we multiply using your method..then arent we using conventional rules...i mean..the same as for integer multiplication...then how do we know this applies for irrational numbers also..I know it does but is there any way to PROVE it

thnx again

What DavidWhitbeck is doing is defining each real number as a particular set of rational numbers (this is the construction of the real numbers via Dedekind Cuts). After you've defined what it means to multiply rational numbers together, you can define what it means to multiply these sets of rational numbers together, and if you do it right, it satisfies all of the properties of the real numbers.

Halls of Ivy is doing something essentially the same, but instead of a specific set of rational numbers, he is instead constructing the real numbers out of the rationals by taking a sequence of rational numbers that converges to the real number. You can them define multiplication of real numbers as just multiplying the 2 sequences term by term. (This is the construction of real numbers as Cauchy sequences)

Both of these constructions are fine. What they do is prove that there are mathematical objects that do act like the real numbers and it gives formal definitions to what it means to multiply them, add them, etc.

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If all you wanted was an intuitive understanding of what it means to multiply 2 real numbers together, then you can just look at geometry: The area of a rectangle gives a good application for multiplying two non-negative real numbers together, and you can use this as a model for what it means to multiply 2 non-negative real numbers (For most of high school and certainly in most of your engineering and physics classes in college, you will essentially use this or some variation as your definition of what it means to multiply 2 real numbers together)

 

[anantchowdhary]

Thanks a lot to all