A Guide from Natural to Imaginary and Infinite Numbers

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"Counting Change" by °Florian

Counting Change Photo by °Florian

Last week’s article introduced N, the set of Natural numbers, the numbers with which we count.

This week resumes with a few functions for Natural numbers, which will extend them beyond their wildest dreams.

Functions and Identity Numbers in Natural Numbers

We had introduced the successor function, using the Greek lower-case “sigma.” In simple terms, σ(i) = i+1 for any Natural number ‘i’ in N.

Since we defined N as the set of all these successors, the result of simply adding ‘1’ to a Natural number stays inside the set N.

Addition and Multiplication as Natural Successor Functions

We can repeat the successor function, much like counting on one’s fingers and toes. The normal addition operation, ‘+’, is just a way of expressing this repeated successor function. Just as “1 = σ(0)” (“one is the successor of zero”), and “2 = σ(σ(0))” (“two is the successor of the successor of zero”), we can write “i = j+k” to show that ‘i’ is the result after “j plus k” successor functions.

Likewise, we can define multiplication as a sequence of identical additions. In both cases, the result remains in the set N. We say “i = j*k” if successively adding ‘j’ to itself ‘k’ times results in the number ‘i’.

Natural Identity Functions

With some more work, we can state that zero is the identity element for addition, since “n + 0 = n”. Basically, if you don’t apply a successor function at all, you keep the original number.

Likewise, one is the identity element for multiplication, since “n * 1 = n”.

Zero is a special number in multiplication, since multiplying anything by zero results in zero.

A New Set, Integers, Arise by Subtracting Natural Numbers

"Negative Temperature" by C Jill Reed

Negative Temperature Image by C Jill Reed

Let’s explore subtraction using a simple analogy with apples. If I have ‘n’ apples, other people can steal some. They cannot steal more from me than I already have, so subtraction in N is limited. “k = i – j” remains in N only if j < i or j = i.

Once we move to accounting principles, we need negative numbers. Suppose I have zero apples, but have an urgent need to keep a doctor away. I borrow one of your apples by promising to give you one in the near future. Then I eat the apple. Now I effectively possess “-1″ apples, because the next one I physically harvest belongs to you. That first apple I pick simply brings me back to zero.

Subtraction opens up negative numbers to our numeric world. The set of Integers consists of Z = {…-2, -1, 0, 1, 2…}.

We could go back to the successor function and define σ(-1) = zero, σ(-2) = -1, and so on. One might prefer to define a “predecessor” function P(n) = n-1, as the inverse of the successor function where σ(n-1) = n. It is difficult to make a set such that the cardinality of that set is a negative number, so it is not a tidy analogue of the successor function.

Perhaps more useful is the notion that “-1 = 0 – 1″. By extension, for any positive integer ‘n’, “-n = 0 – n”.

The point is that introducing the “subtraction” function to N extends that set into the Integers, Z.

Rational Numbers Arise from Divisions Among the Integers

We define division as “a = b/c if b = a * c”. If ‘b’ and ‘c’ are in Z, then b/c is in Z only if ‘b’ is a multiple of ‘c’. Clearly this will not always be true, so we extend Z to the set Q, the rational numbers.

Note that “rational” stems from the word “ratio“, or the proportion of one thing to another.

Rational numbers allow us to express fractions such as 1/2 or -20/3. We can also use the division function to express a fraction as a decimal sequence. 1/2 = 0.5, and -20/3 = -6.666… with an infinitely repeating 6.

Restraining Fractious Fractions

We do not permit division by zero. From the definition “a = b/c if b = a * c”, if we allow ‘c’ to be zero, we see there is no value for ‘a’ that could make ‘b’ anything other than zero. Therefore we cannot divide ‘b’ by zero to determine ‘a’.

Exponents and Roots Lead to Irrational Behaviour and Imaginary Results

We had defined addition and multiplication as extensions of the successor function and of addition, respectively. We can define exponentiation as “a^j = a*a…*a for ‘j’ repetitions of ‘a’, where ‘a’ is a rational number and ‘j’ is a positive integer”.

Let’s also define “a^0 = 1″ (‘a’ to the exponent zero equals one). Let’s omit “zero to the exponent zero”, although one Math Fun Facts page argues that it should be the value one.

Numbers: Image by jprole

Let’s also define “a^(-1) = 1/a” (‘a’ to the exponent “negative one” equals one divided by ‘a’).

We extend negative exponents by saying that “a^(-j) = 1/(a^j)” for all non-zero ‘a’ in Q and all ‘j’ in Z”.

With more work than can be shown in this article, this will eventually extend to “a^b for ‘a’ and ‘b’ in any of the numeric sets”.

Multiplying a number by itself is such a common form of exponentiation that it has its own name: “squaring” the number, shown as “b = a^2 = a * a”. In that case, we say ‘b’ is the “square” of ‘a’.

Once we have the exponent function, we can define a “root” function as its inverse operation. The square root of ‘a’ is the number which, if squared, gives the answer ‘a’. This can be shown as “b = a^(1/2) if b^2 = a”. In fact, the “n-th root” of a number is precisely that number raised to the exponent “1/n”.

Note that there are two square roots for any positive ‘a’, since “-1 * -1 = +1″.

Both exponentiation and taking roots lead to extensions beyond the Rational numbers.

Reality Arrives for Irrational Numbers

Consider the square root of two, and let’s make the case that 2^(1/2) is a rational number. Then “i/j = 2^(1/2) for some ‘i’ and ‘j’ in N and ‘j’ not equal zero”.

To keep it simple, let’s just consider the positive root, rather than the negative root.

If we square both sides of that equation, we have “(i^2)/(j^2) = 2″. Multiply both sides by (j^2) to obtain “(i^2) = 2*(j^2)”.

So (i^2) is an integer, and ‘2’ is one of its factors. Any integer is the product of one or more factors. But whenever we square an integer, the result has exactly two copies of each original factor.

Therefore “(i^2) = 2*2*(k^2)”, for some integer ‘k’. So “2*2*(k^2) = 2*(j^2)”. We can divide both sides by 2, so “2*(k^2) = (j^2)”.

We now can endlessly repeat the claim that, if a square has a factor of ‘2’, it must have a factor of “2X2″, and then keep simplifying the equation endlessly. Since there is no way to resolve this, we conclude there is no rational number i/j that is the square root of 2.

The irony of this was not lost on Pythagoras or his followers. An isosceles right angle triangle has equal sides of ‘1’ and a hypotenuse of “root 2″, by the Pythagorean theorem. Unfortunately, Pythagoras had a religious belief that all numbers are rational. This was discussed in “The Square Root of Two is a Real Irrational Number“.

An irrational number might be expressed by a formula. It cannot be expressed by a repeating decimal expansion, since these can always be resolved into a ratio of integers. (Those who want proof simply need to leave comments with that request!).

The simplest description of R, the set of Real numbers, is that it is the union of all Rational and Irrational numbers.

Imaginary Friends of Real Numbers

"My Imaginary Friend" by stallio with border by Mike DeHaan

My Imaginary Friend: Image by stallio

The union of the sets of Rational numbers, ‘Q’, and irrational numbers, is call the Real number set, ‘R’.

We still need a further extension. There is no real number to satisfy “a = (-1)^(1/2)”, because the result of multiplying a non-zero number by itself is always positive.

But why be so positive all the time? Simply define the principal Imaginary number, ‘i’, as “i = (-1)^(1/2)”. In other words, “i^2 = (i * i) = -1″.

One way to use imaginary numbers involves thinking of it as a coordinate on a Cartesian plane. It is represented either as an ordered pair “(a, b*i)” or a sum “x = a + (b * i)”.

The two square roots of ‘i’ are ( 1/(2^(1/2) )*(1 + i) and ( -1/(2^(1/2) )*(1 + i).

Imaginary numbers have very real uses in physics.

To Infinity and Beyond

We had started with a Natural number as the cardinality of a finite set, and allowed the successor function to keep building larger numbers.

Counting to Infinity, Naturally

What is the cardinality of the Natural numbers? Well, |N| is infinite, larger than any integer can express.

Are there twice as many Integers as Natural numbers? No, since we can make a one-to-one correspondence between these sets. N = {0, 1, 2, 3, 4…} as always. We can list Z = {0, +1, -1, +2, -2…}. Once we get past zero matching to zero, the odd Naturals will match to positive Integers and even Naturals will match to negative Integers. That’s our one-to-one correspondence.

Perhaps there are more Rational numbers than Natural?

No, since there is a one-to-one relationship between Rational and Natural numbers. That topic was covered in “The Paradox of the Infinite Series of Squares Numbers by Galileo“.

Can Infinity be Irrationally Large?

Does the extension from Rational to Irrational numbers create a larger infinity?

Yes, indeed, this was demonstrated in “Cantor Defeated Galileo in the Battle of Infinite Numbers“.

Onward to the Quick Reference Guide

This article and the previous are summarized in “The Definitive Quick Reference Guide to All Types of Numbers.

References:

Hrbacek, K., Jech, T. Introduction to Set Theory. (Third Edition, 1999). Quoted in {Jech, Thomas. “Basic Set Theory”. Stanford Encyclopedia of Philosophy. (2002)}. Accessed Sept. 25, 2011.

Math Fun Facts. “Zero to the Zero Power“. Harvey Mudd College. Accessed Oct. 1, 2011.

Spencer, P. What is the Square Root of i?. University of Toronto Mathematics Network. (April 19, 1999). Accessed Sept. 25, 2011.

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