Expert: Paul Klarreich - 12/19/2006

Question
-----Question-----
When I was first learning math in elementary school we learned the basic math functions: +, -, x, /. But are there basic functions beyond these that are just as fundamental that are not based on them? Such as exponents would require knowledge of multiplication and therefore is not fundamental. Do you know of any such other functions? Thank you.

Sorry I misused the word function when I should have used the word operation. The reason I don't consider exponents to be fundamental is because it uses multiplication. You multiply a number by itself the number of times of the exponent. Are there any operations that don't even particially use the main ones?

Answer
Questioner:  James
Category:  Advanced Math
Private:  no
 
Subject:  Math Functions that are not basic?
Question:  -----Question-----
When I was first learning math in elementary school we learned the basic math functions: +, -, x, /. But are there basic functions beyond these that are just as fundamental that are not based on them? Such as exponents would require knowledge of multiplication and therefore is not fundamental. Do you know of any such other functions? Thank you.

Sorry I misused the word function when I should have used the word operation. The reason I don't consider exponents to be fundamental is because it uses multiplication. You multiply a number by itself the number of times of the exponent. Are there any operations that don't even partially use the main ones?
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Hi, James,

I think we have a semantic discussion here, rather than a mathematical one, but we can leave that for later.  The issue of 'what is a basic function' depends on your mathematical level:

A. Trigonometric functions are considered basic in precalculus mathematics, but, they cannot be computed from their argument just by using the basic OPERATIONS.  [By that, I mean that, although you can compute sin(pi/6) = 1/2 from constructing a right triangle, the 1/2 is not computed FROM the number pi/6.]

B. Functions defined as integrals such as:
[VIEW THIS IN A COURIER FONT, OTHERWISE IT IS NOT READABLE.]

      {x
f(x) = |    exp(-t^2) dt
      }-x

are not computable FROM x using basic operations.  But they are considered basic.

C. Functions such as Bessel functions are considered basic in the study of differential equations, but likewise, etc. etc.

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Now that you are ready for the discussion of 'basic', I'm going to ask you:

Why do you consider multiplication 'basic'?  Doesn't  5*7 mean:

7 + 7 + 7 + 7 + 7

that is, writing down five 7's and adding them up?  If that's the case, then mutliplication is not basic.  

And, since division is based on multiplication, (what should you multiply 7 by to get 35?), division is not basic, either.

And, what does it mean to do 5 + 8?  Doesn't it mean:

A. write down 5 1's:      1  1  1  1  1
B. Now write  7 more 1's: 1  1  1  1  1  1  1
C. Count all the 1's.

So addition is an extension of counting and IT isn't basic.  What really IS basic?  Piling rocks on the floor of the cave?  I don't know.

Are you ready to give up the word 'basic' now?

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And one little comment about your terminology.  This is  off-topic, but I have to say it:

You don't have to apologize to me about careless use of terminology.  Being careful to use the vocabulary correctly will help you a lot in your study of mathematics.

For example, your phrasing regarding powers:

"You multiply a number by itself the number of times of the exponent."

is not accurate and could get you into trouble (as in getting the wrong answer to an example).  When you compute  4^3, it does not mean to multiply 4 by itself 3 times:

4*4 = 16  (multiplied once.)
16*4 = 64 (multiplied twice)
64*4 = 256 (multiplied 3 times)

But  256 is the wrong answer, even though we multiplied three times.  So the exponent (here it is 3) does NOT tell you how many multiplications to do.

It tells you how many FACTORS OF THE BASE TO WRITE.  If you remember that, you will have less trouble in the future.  Have you encountered zero exponents yet?  Have you seen negative exponents yet?  I don't know, but if you have not, you should start using this phrasing all the time, and you will be better off.

So  4^3 means to write  three factors of 4.  And how do you know that you should then multiply them?  Because the word FACTORS means 'numbers being multiplied'.

End of today's sermon.