Assumed Properties of Real Numbers - Axioms

The axiomatic approach in an art or discipline starts with axioms (assumptions) and then combines them with logic through chains of reason to imply further patterns and to obtain results. That provides a logical development of the art and discipline. The first example of this kind of reason was provided by the works of Euclid on Geometry about 300. B. C. It began with a clarification or definition of basic terms, assumed some patterns, and then drew diagrams to represent general patterns and to imply further ones with aid of previously assumed or obtain patterns. The works of Euclid thus included chains of reason which altogether provided a theory of geometry to follow carefully. Chains of reason mimic or employ the human ability to tell and follow stories, one step or moment at a time, one step after another.

These properties of real numbers described below are actually properties of arithmetic operations with real numbers. The real numbers include irrational numbers, rational number, integers, natural numbers and whole numbers. In the modern high school mathematics course designs of the 1955-90, these properties were taken as axioms (assumed properties). That provided a startin point for the algebraic chains of reason, one at a time, one after another, to to imply further patterns and to obtain numerical results.

Chapter 18 in the site Volume 2, Three Skills for Algebra, also describes and illustrate the properties of real numbers given below in a step by step manner.

Algebraic Statement of Arithmetic Properties

To describe the properties rules for changing calculations without changing their results, we introduce four shorthand letters a, b, c and d to stand-in for real numbers (or real quantities). You could pick four different letters if you wish. These letter are actors waiting for roles to be given or assigned. You could pick four different letters if you wish.

You should imagine these rules written with other letters of your choice, when in the calculations you meet, at least one letter a, b, c and d, that has been previously assigned a different role or meaning. In any plot, each actor should have only one role.

The following table describes properties of addition and multiplication which you can use in doing arithmetic or describing arithmetic that could be done. In these laws and properties, the expressions [computation rules] on either side of the equal sign, always give the same result.

In each row of the above table, the first expression or computation rule always gives the same result as the second expression/computation rule,for real numbers or quantities that may replace - be substuted for - the letters a, b and c represent.

Mathematical Practice: In describing a calculation, either expression or computation rule can be replaced by the other, or by a symbol [pronoun] representing the result of either calculation.

Remark. The above properties are assumed and used in doing arithmetic and in changing and manipulating formulas. They are often called the laws of algebra. Do not let that misdirect you. Letters in algebra are place holder for numbers or quantities. So every algebraic formula represents arithmetic or a calculation that might be done. Thus algebraic formulas and manipulations all represent calculations or aithemtic that might be done. Algebraic formulas and expressions, alone or in equations, one side or another, all give computation or calculation rules. Algebra is an arithmetic reflection or shadow.

Right Distributive Axiom For Multiplication

Axiom: If a, b and c are real numbers then

\[ (a+b)c = ac+bc \]

This law is logically equivalent to saying that the two different computation rules \begin{eqnarray*} f(a,b,c) &=&(a+b)c \\ g(a,b,c) &=&ac+bc \end{eqnarray*} give the same result when the values of a, b and c are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a, b and c that the two computation rules give the same result.

Associative Axiom For Addition

Axiom: If a, b and c are real numbers then

\[ (a+b)+c = a+(b+c) \]

This law is logically equivalent to saying that the two different computation rules \begin{eqnarray*} f(a,b,c) &=&(a+b)+c \\ g(a,b,c) &=&a+(b+c) \end{eqnarray*} give the same result when the values of a, b and c are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a, b and c that the two computation rules give the same result.

Commutative Axiom For Addition

Axiom: If a and b are real numbers then

\[ a+b = b+a \]

This law is logically equivalent to saying that the two different computation rules \begin{eqnarray*} f(a,b) &=& a+b \\ g(a,b) &=&b+a \end{eqnarray*} give the same result when the values of a and b are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a and b that the two computation rules give the same result.

Associative Axiom For Multiplication

Axiom: If a, b and c are real numbers then

\[ (ab)c = a(bc) \]

This law is logically equivalent to saying that the two different computation rules \begin{eqnarray*} f(a,b,c) &=&(ab)c \\ g(a,b,c) &=&a(bc) \end{eqnarray*} give the same result when the values of a, b and c are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a, b and c that the two computation rules give the same result.

Commutative Axiom For Multiplication

Axiom: If a and b are real numbers then

\[ ab = ba \]

This law is logically equivalent to saying that the two different computation rules \begin{eqnarray*} f(a,b) &=& ab \\ g(a,b) &=&ba \end{eqnarray*} give the same result when the values of a and b are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a and b that the two computation rules give the same result.

Left Distributive Axiom For Multiplication

Axiom: If a, b and c are real numbers then

\[ c(a+b) = ca+cb \]

This distributive axiom follows from the others. Let show how. The derivation of the left distributive law below from earlier ones provides another illustration of the algebraic way of writing and reasoning in mathematics.

\begin{eqnarray*} c(a+b) &=& (a+b)c \\ &=& ac+bc \\ &=& ca+cb \end{eqnarray*}

Each expression in the previous chain of equalities gives a computation rule equivalent to that provided by the expression before or after it. That implies \begin{eqnarray*} f(a,b,c) &=&c(a+b) \\ g(a,b,c) &=&ca+cb \end{eqnarray*}

give the same result when the values of a, b and c are real numbers.

Exercise: Write a computer or calculator programs to evaluate f and g, and then observe for randomly choosen values of a, b and c that the two computation rules give the same result.

Additive Identity For Addition

Axiom: If a is a real numbers then there a number 0 such

\[ a+0 = 0+ a = a \]

This law is logically equivalent to saying that three computation rules \begin{eqnarray*} f(a) &=& a+0 \\ g(a) &=&0+a \\ I(a) &=&a \end{eqnarray*} give the same result when the values of a and b are real numbers.

This axiom or property is used in arithmetic and in algebra to transform one expression a into another with the aid of the other properties or axioms. Examples will illustrate this further.

Exercise: Write a computer or calculator programs to evaluate f g and I, and then observe for randomly choosen values of a that the above computation rules give the same result.

Additive Identity For Multiplication

Axiom: If a is a real numbers then there a number 1 such

\[ a \times 1 = 1 \times a = a \]

This law is logically equivalent to saying that three computation rules \begin{eqnarray*} f(a) &=& a\times 1 \\ g(a) &=&1\times a \\ I(a) &=&a \end{eqnarray*} give the same result when the values of a is a real numbers.

This axiom or property is used in arithmetic and in algebra to transform one expression a into another with the aid of the other properties or axioms. Examples will illustrate this further.

Exercise: Write a computer or calculator programs to evaluate f g and I, and then observe for randomly choosen values of a that the above computation rules give the same result.

Note for Course Designers

The Algebra Starter Lesson appendices

1. A Origins of Counting and Figuring Methods
2. B Real Numbers Extrinsic Development

The above employment of computation rules is intended to make comprehension of the axioms clearer. The the notion that a function is given by a set of equivalent computation rules departs from pure set theory perspective, but does so in a manner that aids an operational command of the concept.

Numerical Exercises

Use the axioms given above to identify which pairs of arithmetic expressions below will give the same result. Calculator use in advance to identify which arithmetic expressions give the same result would be defeat the use and illustrate the axioms purpose here.

Computation Rule Exercises

$f(x,y,z)= x(y+z)+x \cdot z$

$h(x,y,z)= x y+x z +2x^2 \cdot y$

$t(x,y,z)= +x \cdot z + (y+z)x$

$u(x,y,z)= (y+z) z +2x^2 \cdot y$

$v(x,y,z)= (x y+x z)2x^2 \cdot y$

$u(x,y,z) = ( 2 yx^3+x^3 y^2) z$

$p(x,y,z) = ( 2 yx^3+2x^3 y^2) z$

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science