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: (x n)(x p) = x n+p
• Some First Consequences of the law of exponents:
• (x n) p = x np.
(x n)/(x p) = x n-p
• Extending the notation
• 1 as an exponent, x1 = x
• 0 as an exponent, x0 = 1.
• negative exponents, x-n = 1/(x n)
• inverses as exponents,   x(1/n) =   nx
• fractional (or decimal) exponents,  xn/p =  ( px) n
• the exponential function,exp(x, y) = x y

#### 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.

• x1 = x
• For any positive integer n, it makes sense to define x-n = 1/x n .   This is just the extension of the law of exponents to include negative integers for we have  xp x-n = x p1/x n = x p/x n =  x p-n  = x p+(-n) .   The key step here is the realization that x p/x n =  x p-n   follows by cancellation.  For example
• x 5 x -3  =   x x x x x/x x x  =   x x x x x  /x x x  =  x x = x 2
• x 3 x -5  =   x x x / x x x x x  =   x x x / x x x x x  =  1 / x x  = x -2.
• x0 = 1,  this is just the special case where for any positive integer n we have x0 = xn-n = xn / xn = 1.
• (x n) p = x np, because  x n equals x multiplied by itself n times, while (x n) p equals x multiplied by itself p times and so(x n) p  equals x multiplied by itself np times.
• The above rule suggests the following rule for extending the law of exponents to rational numbers by taking  x(1/n) = nx   .  This makes sense since n denotes the positive number, y, such that yn = x (i.e., y multiplied by itself n times gives x) , but since (1/n)n = 1,  the preceding rule gives us ( x(1/n) )n = x, just as desired.
•  Generalizing we get: xn/p =  ( px) n as the meaning of an arbitrary fractional exponent n/p.

•

#### 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

• For n>1, enalways starts with  2
• For n>4en always starts with 2.7
• For n>5, en always starts with 2.71
• etc.
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
• 2.718281828459045235360287 (e  approximated to  twenty-five decimal places).
• 2.7182818284590452353602874713526624977572470937000 ( e approximated to fifty decimal places).
• 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427 (e approximated to  one hundred places).
• Note the difference between the four digits at the end of the fifty-place approximation  and the corresponding digits in the 100-place approximation.  Why do you think they are different?

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