The Definitive Quick Reference Guide to All Types of Numbers

Calkin–Wilf sequence of natural numbers: Image by David Eppstein

Background to this Reference Guide to Numbers

Several previous articles introduced a variety of numbers, including Natural Numbers, Integers, Rational Numbers, Real Numbers, Imaginary Numbers, Irrational Numbers, Infinite Numbers, and others. This quick reference guide is intended to provide additional information and a summary of the main types of numbers.

The Sets of Numbers

The below list is organized so that each set of numbers in the list includes all the sets listed above it. For example, Integers, second on the list, includes all Natural Numbers.  The exceptions are noted briefly within the list. Further notes are listed below.

  • N = Natural Numbers = {0, 1, 2, …}. These are also known as the “counting numbers”. Sometimes zero is excluded from this set, but included in the “Whole” numbers.
  • Z = Integers = {…, -2, -1, 0, 1, 2, …}. This set is the union of Natural Numbers and their counterparts, negative numbers.
  • Q = Rational Numbers = {i/j} for all ‘i’ and ‘j’ in the Integers, excluding any fraction with j=0.

There is a lot to be said in the section below.

  • Irrational Numbers are all “Real” numbers that cannot be expressed as a ratio of integers. This is a poor definition, because it refers forward to the next set. The previous article explained why the square root of two is irrational, but did not explain the whole set. See the notes below, both for “square root” and especially for Irrational numbers. The above sets set are not included in this one.
  • R = Real Numbers consist of the union of all Rational and Irrational numbers.
  • Imaginary Numbers are required to solve problems involving the square root of negative numbers.
  • Infinite numbers express a cardinality for infinite sets of numbers. The above sets set are not included in this one.

<h2″>Natural Numbers

Start by defining zero as the cardinality of an empty set. Then define a successor function that makes a new set by including the previous set as a new element, along with any elements in the previous set. Define a “greater than” relationship: when you make a one-to-one relationship for the members of two sets, then the cardinality of the set with extra members is greater than that of the other set.

The addition operation is defined from the successor function. For natural numbers ‘i’ and ‘j’, the value “i + j” applies the successor function to zero, first ‘i’ times and then ‘j’ times.

The multiplication operation, “i * j”, is found by repeating “i + i…” for ‘j’ occurrences of ‘i’.

Zero is the unity number for addition, making no change to its partner. “i + 0 = i”. One  is the unity number for multiplication: “i * 1 = i”.

Please refer to “A Quick Reference Guide to the Set of Natural Numbers” for more details.

Integers

Natural numbers are not “closed” under subtraction, since “0 – 1 = -1″. It is rare to see the set of negative integers stand alone as a set, but it is convenient to use that as a stepping-stone to define the Integers.

Define the subtraction operation from a predecessor function, the inverse of the successor function. If the successor of zero is one, then the predecessor of one is zero. What is the predecessor of zero? “Negative one” is not a number for counting, but accountants use it to indicate a debt.

The “Guide from Natural to Imaginary and Infinite Numbers” provides more details about Integers, as well as the rest of these other sets of numbers.

Rational Numbers

Literally, a rational number is a “ratio” of integers. Define the division operation as the inverse of multiplication: “z = x/y means that x = z*y”.

Irrational Numbers

Irrational numbers were not fully explained in “The Square Root of Two is a Real Irrational Number“, which did demonstrate that the square root of two is irrational. As the multiplication operation is repeated addition, so squaring a number is the process of multiplying a number by itself: “x^2 = x*x”. The square root is the inverse of squaring: “y = x^(1/2) if x = y*y”.

Another group of irrational numbers are the transcendentals, such as pi (‘π’=3.14159…); phi (‘φ’, the Golden Ratio, =  (1 + 5^(1/2) ) / 2) = 1.618…), and ‘e’ = 2.718281828…, the base of natural logarithms.

The Fraktur symbol is used to denote the imaginary portion of a complex number.

Irrational numbers were not well covered in the previous article. Richard Dedekind pursued Irrational numbers by seeking continuity in the “greater than” relationship.

Around 1872, Dedekind showed that one could define a number line, with the origin at zero and an arbitrary distance of one unit in a particular direction. He then demonstrated that every Rational number could be found on that number line, and the “greater than” relationship would apply. For example, the point 1/2 is found where one divides the number line between {x : 1/2 > x} and {x : x > 1/2}.

However, we should also be able to “cut” the number line at {x : 2 > x^2} and {x : x^2 > 2}; that is, at the square root of two. Since this is an irrational number, a number “line” composed solely of Rational numbers is not continuous. It has gaps.

We can say that the Irrational numbers fill in the gaps between Rational numbers on the number line.

Real Numbers

The set of Real numbers is the union of Rational and Irrational numbers. Despite the above paragraph dealing with Irrational numbers, in my opinion it is easier to begin thinking about Real numbers as the continuous number line, and then look back to divide the points into Rational or Irrational numbers.

Imaginary and Complex Numbers

After defining the square root function as “y = x^(1/2) if x = y*y”, there is a problem with closure of Real numbers. Since “(-1)^2 = (-1) * (-1) = (+1) = 1″, it is obvious that “y = (-1)^(1/2)” has no solution in the Real numbers.

"Polar Coordinates" by Mike DeHaan

Polar Coordinates by Mike DeHaan

Define “i = (-1)^(1/2)” as the foundation for Imaginary numbers, so “i^2 = -1″. Imaginary numbers are all {y*i} where ‘y’ is any Real number and ‘i’ is
the square root of negative one. Would you count Real numbers as a subset
of the Imaginary, or are they completely disjoint sets?
Complex numbers are all {x+y*i}, where both ‘x’ and ‘y’ are Real. We would say that “The Real component is ‘x’ and ‘y’ is the Imaginary component” of this number.

View this representation of a Complex nuumber as an (x, y) point on the Cartesian plane.

Yet another approach using the Cartesian plane is to plot the (x,y) point as a distance ‘r’ at an angle “theta” (‘θ’). The center is (0,0) and the angle is counter-clockwise from the positive x-axis. “(r, θ)” means (x = r*cos(θ), y = r*sin(θ) ), using the trigonometric functions “cos”=cosine and “sin”=sine.

Infinite Numbers

The first Infinite number is the cardinality of the Natural numbers. Since Integers, as well as Rational numbers, can be placed into a one-to-one relationship with the Natural numbers, they all have the same infinite cardinality.

Cantor Defeated Galileo in the Battle of Infinite Numbers” demonstrated that the cardinality of Real numbers is greater than the cardinality of the Natural numbers. In other words, one can always create a new Real number that was not”counted” by the Natural numbers.

Aleph numbers represent the size of infinite sets. Image by Renaud Camus

Aleph numbers represent the size of infinite sets. Image by Renaud Camus

According to Jeff Suzuki’s ”Mathematicians and Other Oddities of Nature“, the cardinality of Natural numbers is called “aleph-null”. (“Aleph” is the first letter of the Hebrew alphabet; see the image). There are higher infinities, called “aleph-one”, “aleph-two”, and so on. The cardinality of the Real numbers is denoted ‘C‘ (for “Continuum”). However, it has been proven that it cannot be proven that C is precisely “aleph-one”.

External References:
Reck, E. Stanford University Encyclopedia of Philosophy. Dedekind’s Contributions to the Foundations of Mathematics. (rev. Sept. 6, 2011). Accessed Oct. 9, 2011.

Stapel, E. Complex Numbers: Introduction. Purplemath. (2000-2011). Accessed Oct. 17, 2011.”

Suzuki, J. Boston University. Mathematicians and Other Oddities of Nature. Accessed Oct. 9, 2011.

Talbert, R. Casting Out Nines. How to memorize the value of e to 15 decimal places. (Feb. 9, 2010). Accessed Oct. 9, 2011.

Weisstein, E. W. MathWorld–A Wolfram Web Resource. Polar Coordinates. Accessed Oct. 9, 2011.

© Copyright 2011 Mike DeHaan, All rights Reserved. Written For: Decoded Science