Properties of Real Numbers
algebraically described
Rule-based reasoning is used in the changing of formulas
and equations. Somewhat flexible rules say how or what is permitted. The
flexible rules in algebra can be applied one at a time or one after another to
arrive at formulas and equations or to draw conclusions on or from them (the
formulas and equations, that is). But understanding the rules requires the
algebraic way of writing and thinking to be well understood beforehand, else
they, the rules, will not make sense.
The numbers first met in arithmetic are called real numbers. Each of these
numbers can be written in decimal notation (with a sign perhaps) or as a
fraction. More precisely, there are various kinds of real numbers:
- the whole numbers 1, 2, 3, 4, 5, ....
- the number zero: 0.
- integers: 0, ±1, ±2,
±3, ±4, ±5,
±6, ....
- rational numbers, that is, fractions or ratios, ±[(p)/(q)]
in which p and q stand for whole numbers with q ¹
0. Each fraction has a periodic or repeating decimal expansion.
- irrational numbers p and Ö2
etc. Each irrational number is given by a non-repeating, non-periodic,
decimal expansion.
The set of real numbers consists of all these numbers. The term real number is a
bit distracting. When you see it, just remember this: real numbers can be
written as decimal expansions or as fractions, with plus and negative signs in
front as prefixes. Here +2 and 2 are taken to denote the same number
Real quantities are given by a real number times a unit of
measurement. In elementary school and in high school, we should meet examples
of calculations involving both numbers and quantities. The rules of arithmetic
(given below) also apply to real quantities.
3 Arithmetic Rules
- What They Do. The rules of arithmetic say when the order of
operations can be changed in a first calculation, so that we obtain a second
calculation which gives the same result as the first. These rules apply to
arithmetic involving real numbers and/or real quantities.3
- The order of arithmetic operations, suggested by parentheses, matters in
some calculations, but there is some flexibility. In some but not all, we
can change the order in which arithmetic is done without changing the
arithmetic result. The properties of arithmetic (rules) given below say how
this can be done.
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). The use of these
letters is a tradition. Other letters could be used. Sometimes it is convenient
to describe or rewrite these rules or properties using other letters.4
You could pick four different letters if you wish.
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 on either
side of the equal sign, always give the same result.
|
Properties of Addition and Multiplication
|
| First expression |
= |
Second expression |
name of the property (or rule) |
|
(a+b)+c = a+(b+c)
|
associative law for addition |
|
(ab)c = a(bc)
|
associative law for multiplication |
|
(a+b)c = ac+bc
|
(right) distributive law |
|
c(a+b) = ca+cb
|
(left) distributive law |
|
a+b = b+a
|
commutative law of addition |
|
ab = ba
|
commutative law for multiplication |
|
a+0 = a
|
additive identity: the effect of adding zero |
|
a·1 = a
|
multiplicative identity: the effect of multiplying by one. |
In each row of the above table, the first expression always gives the same
result as the second expression, no matter what real numbers or quantities the
letters a, b and c represent. In describing a calculation,
either expression can be replaced by the other, or a symbol (pronoun)
representing the result of either calculation.
The above rules only involve addition and multiplication. We will talk next
about the above properties and rules and about how they are used, next. How to
apply these rules to expressions involving subtraction or division will also be
described later.
Reminder. The product a×b is also written as a·b
or as ab. Which notation is used to signal multiplication is a matter
of taste and convenience. When the times symbol × might be confused with the
letter x, remember to use the dot · instead, write a·b
or ab.
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. This author prefers to call them laws or properties for
arithmetic.
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