More About Exponential Notation

Outline for Exponential Notation








We introduced exponential notation,  in the material on Arithmetic Expressions, as a way of abbreviating certain arithmetic expressions  For example,  we used exponentiation notation to abbreviate 555 as 53 .   Later, in the material on algebraic expressions we used exponential notation on variables,  writing x as an abbreviation for xxx.  In those situations the exponent was always a positive integer.  We will now go on to discuss how to interpret exponential notation in the more general situation where the exponent is allowed to be any rational number.
 
 
 

The Law of Exponents:

What is the result of multiplying 53 by 54 ?

Well, since 53 = 555 and 54 = 5555 we must have

(554) = (555)(5555) = 5555555 =  57 = 53+4 .
This example suggests that for every  real number x  and all positive integers n and p  we might have the general rule
(x n)(x p) = x n+p .
If you try other examples you will see that this rule always holds.  This rule is called the law of exponents.
 

Some First Consequences of the law of exponents:

We can use the law of exponents to derive some other useful rules.
 
 

 

Extending the notation to arbitrary real numbers as exponents.

 

 
 
 
 
 
 
 
 
 
 
 

Exponents and Logarithms

 Recall that, for any natural number n, that the factorial of n, written as  n!,is the number  n! = n(n-1).(n-2) ... 21. So, for example,  3! = 321 = 64!  = 4321 = 24, and 5! = 5321 = 120.

Consider the infinitely long expression

1 + (x / 1) + (x 2/ 2!) + (x 3 / 3!) + ... +( x n /n!) + ...

This expression is called the MacLaurin series for the exponential function.

If we, say, plug in the value 1 for x we get the infinitely long arithmetic expression

1 + (1 / 1) + (1 / 2!) + (1 / 3!) + ... +( 1 /n!) + ...

Let  ei denote the result of evaluating the first i terms of this expression .  We get a sequence of numbers

    e1 = 1
    e2 = 1 + 1 = 2
    e3 = 1 + 1 + 1/2! =   1 + 1 + 1/2 = 2.5
    e4 = 1 + 1 + 1/2! + 1/3 =  1 + 1 + 1/2 + 1/6 =  2.666666....
    e5 = 1 + 1 + 1/2! + 1/3! + 1/4! =  1 + 1 + 1/2 + 1/6 + 1/24 = 2.70833...
    e6 = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! =  1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.7166...
    ...
    e10 = 2.7182815255731922399....
    ...
    e15 = 2.7182818284582297479...
    ...
    e20 = 2.7182818284590452349...
    ...
    en+1 =  en + 1/(n-1)!

If you inspect this sequence of numbers you will see that as you go up the sequence the numbers "become more and more alike" -- that is
 

This suggests that  the sequence  is converging on (becoming increasingly and arbitrarily close to) to some particular real number.  This is indeed the case -- the number is question is called e.  The real number e is an  infinite non repeating decimal, which mean that we can not write it down completely.   But by making use of the above series we can write it down to any desired degree of accuracy.  For example


If  we evaluate the expression 1 + (x / 1) + (x 2/ 2!) + (x 3 / 3!) + ... +( x n /n!) + ... with x = 2,  we will get an infinite sequence of numbers which converges to 7.389056098930650227...   =  ee =  e2 .   In fact,  if we evaluate the expression  with x a rational number,  a/b,  then the result will be ea/b.   Finally (for now), if we evaluate the expression at any real number r then the result is er.

We can also show that if r is any positive real number   that there exists a number, let's denote it as log(r), such that elog(r) = r.   Then, given any two real numbers,  r and s we can compute   rs  by  the rule  rs = (elog(r))s = e s log(r) .   The number log(r) is called the natural logarithm of r