The Elegance of Exponents
An exponent is generally expressed as follows
In simple parlance we say that the original value of a “base” is operated upon by an “exponent” transforming it into some other value.
In subsequent sections, we will restrict variables to the sets described in the “APPENDIX: Infinite Number Sets”. Initially we require that our base “a” is a real positive number. This can be expressed formally for our “base” as
Early in the development of mathematics, exponents were invented as elegant shorthand for what would otherwise be a cumbersome series of repeated multiplications. In the simplest case, we assume that our exponents, “m” and “n” are restricted to the positive integers so that
So that the addition of a minus sign means that
Using integers, an exponent indicates the number of times a base is multiplied by itself.
* a * a
Then the expression is shorthand meaning the base “a” is to be multiplied by itself “n” times. This leads us to two important relations. The first case is when two terms with the same base, “a”, but different exponents are multiplied. In this case, the exponents are added
The second case is when the quantity of a base with an exponent has a second exponent. In this case, the exponents are multiplied.
As for example
If on the other hand we were to divide bases to various powers we might have
Or for special cases
RATIONAL NUMBER EXPONENTS
Rational numbers are those which can be expressed as a fraction with an integer for the numerator (number on top) divided by a non-zero integer for the denominator (number on the bottom. If “r” and “s” are integers and s is not zero, we can express this as
Then the fraction “t” is called a “rational” number
An advantage of rational numbers is that we can use an exponent to represent a square root or indeed any arbitrary root. Recall that the square root of “a”, is that number which when multiplied by itself gives a.
But since exponents add for multiplications with the same base “a”, we can write
Solving for “t” we have
We might then write, for example
We can generalize this result to note that a1/s is simply the “s-th” root of “a”, or the number which when multiplied by itself “s” times gives “a”.
Or more simply we simply have the “s-th” root of “a”, as follows:
Continuing the analogy, we can use arbitrary fractions for the exponent, as given below:
or that a( r/s ) is the “s-th” root of the base “a” multiplied by itself “r” times.
And as before if the rational fraction for the exponent is negative, we can write
IRRATIONAL NUMBER EXPONENTS
So far we have considered rational fractions, t = (r/s), where r and s are integers and s is not zero. There are many other numbers, however, which cannot be expressed as any rational fraction and are therefore called “irrational.”
The most famous irrational number is the square root of two. Sometime around the 6th century B.C. the Pythagorean Greeks discovered that any rational number fraction which might be equal to the square root of two had to have a numerator and a denominator both of which were infinitely divisible by two. Since no integer has these properties, the square root of two cannot be expressed as a rational fraction.
Please note that all rational numbers, or fractions, when expressed as decimal numbers eventually have strings of digits that repeat forever. Irrational numbers never have repeating sequences of digits.
Nevertheless for any irrational number, we can always find some rational fraction which approximates it to any degree of precision for which we have the patience. That is to say that for any irrational number “q”, we can approximate it in the limit
Then as consequence, we can also write
One way to approximate “q” is to write it as a decimal for as many digits as are desired; and then substitute an integer divided by some power of ten.
REAL NUMBER EXPONENTS
The rational fractions and irrational numbers together comprise the number set called the “real” numbers, . And we have previously described exponents for both of these subsets which together constitute the entirety of the set
IMAGINARY NUMBER EXPONENTS
An interesting property of any real number, whether positive or negative is that if you multiply that number by itself, the result is always positive. This means there is no “real” number that could be the square root of any negative “real” number. This is a direct consequence of the fundamental assumption of algebra called the “Distributive Law” or
For a variety of calculations which are essential to the development of modern technology, we need the square roots of negative numbers. The solution was to invent a non-physical quantity called “i” which is defined as the square root of minus one.
The set of all imaginary numbers, , is simply every real number, “x”, multiplied by the weird constant “i”, as follows:
where simply means “for all”. Around 1740 A.D., Swiss mathematician Euler discovered an elegant formula describing imaginary exponents. The starting point is to define a function using trigonometric functions and the constant “i” as follows
Taking the derivative with respect to x, we get
Rearranging terms and integrating both sides with respect to “x” yields
The result of the term on the left hand side is the natural logarithm, ln(z), to base “e” which is a constant approximately equal to 2.718281828…
The result is
And finally, we note the imaginary exponent has real and imaginary parts, as
This can be visualized as follows
Values along the “imaginary” axis are simply real numbers multiplied by “i”. The angle x is expressed in radians. Every point in the “complex” plane is thus some value of r*z(x) which has a real and imaginary part. If r = 1, then every possible result for “ eix “ lies on the circumference of the corresponding circle.
Some bizarre examples relating fundamental numbers of nature include the following
What exactly is the significance of these relations is still to be determined.
Connecting imaginary numbers with trigonometry is de Moivre’s formula when and , then
Likewise we can write the trigonometric functions as
COMPLEX NUMBER EXPONENTS
A “complex” number is defined as simply the sum of a real number and an imaginary number so for
EXPONENTS USING MATRICES
Matrices can also be raised to powers. For instance we might have an arbitrary matrix “A” with “m” rows and “n” columns, then
If this matrix is square, “Am x m“, only then we can define a Natural number exponent “p” as follows
or in general
where the matrix “A” is simply multiplied by itself “p” times. And for the special case
where the “Identity” matrix “I” has ones along the diagonal but zeros everywhere else. This is not to be confused with the “Zero” matrix, Z”, which has zeros everywhere as
Also note that the inverse matrix “A-1“ has a specifically different meaning than we might expect as follows
But a matrix can also be used as an exponent (so that “eA “ is itself a matrix) as follows
Note that in the special case where the matrix has only diagonal non-zero elements as
APPENDIX: Infinite Number Sets
Please note that we will use the following symbols to represent different well-known infinite number sets.
- Natural or counting numbers
- Rational numbers
- Real numbers
- Imaginary numbers
- Complex numbers
where “∀” simply means “for all”, “∈” means ”is a member of the set”, and “|” means “such that”.