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Numbers and Their Application - Lesson 2

Numbers: The Integers

Lesson Overview

The Development of Mathematics via Axioms, Definitions, and Proof

An axiom is a statement assumed to be true.

Postulate is another word for axiom.

Axioms and logical reasoning together enable mathematicians to prove things. In this section we will present and discuss certain axioms from which all the properties of the natural numbers may be proved. Later lessons will develop the concept of logical reasoning and proof. First, we will present groups of axioms to help us understand the different number systems we will encounter.

Undefined words in today's lesson include the following: equal, successor, and number. The terms addition, multiplication, subtraction, and division will also not be rigorously defined, but must satisfy the group and field axioms presented in Lesson 7 and lesson 13. You were taught rudimentary algorithms in grade school which we will review very briefly.

Natural or Counting Numbers and Whole Numbers

The natural or counting numbers are the familiar set: {1, 2, 3, 4, 5, ...}

The symbol ... (often read as dot dot dot) is often abbreviated etc., which is an abbreviation for the Latin term et cetera meaning and so forth. There is actually no uniform agreement as to whether or not zero (0) is a natural number. Popular usage indicates that it is not, whereas books on number theory will often define it to be one! Computer scientists and some popular programming languages such as C and C++ also often treat it as a counting number. The difference can be summarized by where we point or index (cf your index finger). (cf is an abbreviation for the Latin confer meaning compare.)
Most books define whole numbers as the union of the counting numbers with zero.

Zero and One Indexing

[Finger pointing to the right] Zero Indexing acknowledges zero as the number we start counting with.
[Finger pointing to the right] One Indexing acknowledges one as the number we start counting with.

In this class we will be flexible, but try to specify when zero indexing is to be used.

The symbol N is used to denote the set of natural numbers.

Infinity

The set of natural numbers is an infinite set. There is always a next larger number. The perhaps misguided concept of "biggest number" is usually conveyed by the term infinity and symbol [infinity]. This concept and the concept of continuity were the root cause of several ancient Greek paradoxes which we will explore further in Lesson 13.

Peano Axioms and Mathematical Induction

  1. 1 is a member of the set N.
  2. If n is a member of N, then n+1 belongs to N (where n+1 is the "successor" of n).
  3. 1 is not the successor of any element in N.
  4. If n+1 = m+1, then n = m.
  5. A subset of N which contains 1, and which contains n+1 whenever it contains n, must equal N.

In general, we don't emphasize the above axioms in this class, but they are presented here to assure you the natural numbers exist (just in case you had any doubt).

Axiom 5 above is the basis for mathematical induction which will be developed later.

Addition and Triangular Numbers

When we add two numbers together, they are termed addends. The result is termed the sum.

An interesting subset of the natural numbers generated by addition are called Triangular Numbers. These are so called because these are the total number of dots, if we arrange the dots in a triangle with one additional dot in each layer.

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The triangular numbers thus are: 0, 1, 3, 6, 10, 15, 21, .... (Not everyone considers 0 to be triangular.)

The following example has a rich history dating back to the early childhood of Gauss. To keep his class busy for a long time, the teacher told them to add the counting numbers up to one hundred. Gauss finished very quickly thus revealing his early genius. This is what he did:

T100 = (1 +100) + (2+99) + (3 + 98) + ... + (50 + 51)
= 101• 50
= 101• 100/2

Note how the equal signs are aligned vertically, a form we will strongly encourage to reduce mistakes.

This can be generalized to: [The sum over i from 1 to n of i =n(n+1)/2] where mathematicians use the capital Greek letter [Sigma] (sigma) to represent summation. One of your teachers has a particular fondness for this symbol since the first computer he had much access to had that nickname.

Fibonacci Numbers

Another way to add numbers together generates the Fibonacci Numbers. These numbers are named after a great 13th century mathematician who studied them.

Fibonacci Numbers, represented here by Fi, can be defined as follows.
Let F0=0 and F1=1. For all other Fi, let Fn+1 = Fn-1 + Fn.

This definition is recursive, i.e. each term is defined in terms of the previous two. The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (Not everyone considers 0 to be a Fibonacci number.)

Factorials

When we multiply a multiplicand by a multiplier, the result is termed the product.

Factorials can be defined recursively as n! = n•(n-1)! where 1!=1.
By definition, 0!=1. (Don't ask, it just works best!)

For example, 5! = 5•4•3•2•1 = 5•4•3•2 = 5•4•6 = 5•24 = 120.

In general, [n!=pi product over i=1 to n of i] The symbol [Pi] is the capital Greek letter pi and represents product. The expression is termed a pi product.

Subtraction and the Rest of the Integers

Early in life, most of us encounter negative numbers, for example, when something costs more than what we have. Perhaps, we are able to get an advance on our allowance and thus encounter debt.

When you subtract a subtrahend from a minuend, the result is termed the difference.
The integers are the counting numbers together with their opposites and zero.

Opposite in this case refers to the concept of additive inverse (a field axiom). It would seem that we have doubled the size of the number system, but in actuality it is still a countably infinite set.

The symbol Z is used to denote the set of integers.
It comes from the German word zahlen, meaning to count.

Even: An integer is even if it is an integer multiple of 2.

Odd: An integer is odd if it is not an integer multiple of 2.

Hence, the even numbers are 0, ±2, ±4, ... and the odd numbers are ±1, ±3, ±5,... Zero is even.

Integer Division or Division with Remainder, Modulo, Congruence

Although division will be presented again later, a special form will be introduced here. Often the remainder obtained in a division is more important than the quotient.

When a dividend is divided by a divisor, the results are termed the quotient and remainder, where quotient is the number of times the divisor went into the dividend and the remainder is how many were left over.

Quotient    R  Remainder
When doing long division, it looks like this:    Divisor /Dividend

The concept of even and odd introduced above can be expressed as whether the remainder was 0 or 1 when divided by 2. This can be expressed as 0 mod 2 or 1 mod 2 where mod is an abbreviation for the Latin term modulo meaning a small measure. The same syntax is often used to ask the question: What is 121 mod 2? Answer: 121 is 1 mod 2, or an odd number. We also say, 121[equivalent to]1 (mod 2) Where [equivalent to] is read equivalent to (and should be about the size of a normal equal size). A later homework problem will extend this concept to your every day experience such as telling time.

Modulo is the remainder when dividing by a divisor.

Numbers which have the same remainder when divided by another are termed congruent. Congruence will have other uses in geometry to indicate two objects have both the same shape and measure.

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