Chapter 18 Rules for Algebra

or Arithmetic Rules and Patterns (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 new formulas and equations. But understanding the rules requires the algebraic way of writing and thinking to be well understood. This chapter aims to  make the algebraic description of the properties of real numbers understandable and useable.  For many students, the algebraic shorthand description of numerical properties is gibberish - ouch. Explicit and deliberate rationalization or explanation is needed. Providing that is the aim below and in newer site algebra 12 Starter Steps, the next two included.

• 4. Computation Rules and Function Notation
• 7. Axioms, Logic and Equivalent Equations/Computation Rules

If these two are not to your liking, study the others. Sit steps, this and further site chapters and offer less steep paths to learn and teach. Less steep implies easier, but it also implies longer - the cost of being less steep. Good luck.

1  Order of Operations

Parentheses are (often) used to show the order in which arithmetic (+, -, × and ×) is done in a calculation. The order can sometimes be changed without changing the result. Rules or properties of arithmetic say when. These rules and properties say how to move the parentheses about, and how to omit them, without changing the result obtained. Here the arithmetic may change, but the result does not.
In this section, we talk about the use of parentheses in the description of calculations that are or could be done. The rules of arithmetic for shifting or omitting parentheses state when two would-be calculations should give the same result are described in the following sections.
In arithmetic, the order in which the arithmetic is done may change the result. So some caution is required. In describing calculations we also need to give the order in which the additions, subtractions, multiplications and divisions can be correctly done. The order is based on the following conventions:

• expressions within a pair of parentheses (¼) are to be computed before those outside. So the stuff ¼, whatever it is, within a pair of innermost parenthesis are done before those outside.
• without parentheses to show what calculation is to be done, multiplications and divisions are to be done before additions and subtractions. Multiplication and division are said to have a higher priority.

Departing or changing the order in which arithmetic is done could give an incorrect answer. Here are some more examples which show that the order of operations sometimes affects results. Your problem is to know when.

1. The expression 17-(10-3) gives 17-7 = 10 but (17-10)-3 gives 7-3 = 4.
2. The expression ([4/5] ×[5/16])×[2/3] gives

   / | \ 4 5 · 16 5 \ | / ÷ 2 3 = 4×16 5×5 · 2 3 = 64 25 · 2 3 = 128 75

   This is different from

   4 5 × / | \ 5 16 · 2 3 \  | / = 4 5 × / | \ 5 24 \ | / = 4 5 · 24 5 = 96 25

3. The expression (5×6)×2 = ([5/6])×2 = [5/12] but 5 ×(6×2) = 5×3 = [5/3]. The parentheses cannot be omitted.
4. (8-5)-2 = 3-2 = 1 while 8-(5-2) = 8-3 = 5. So the parentheses are important.
5. But (5·4) ·3 and 5 ·(4 ·3) both give the same result.

Sometimes the order in which arithmetic is done affects the result. In this case, parentheses and conventions are needed to say what is done first. So unless you know a rule which says the order indicated by parentheses and the priorities assigned to arithmetic operations (+, -, ×, ×) can be changed, you should be very careful. When in doubt, don't.

In teaching, I had respect for the student who would identify in his or her arithmetic (or reasoning) what was uncertain. That was a sign of careful thinking. I tried not to reward students who tried to hide their guesses. In learning, once a student has identified the limits and uncertainties in his or her knowledge, that student is ready and able to learn more.

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2  Working with Formulas
Changing Calculations

Changing a formula for a number (or quantity) can reduce or lessen the amount of arithmetic needed to calculate it.¹²

²Computers can be told (programmed) to calculate results for us. One method to compute a result may require ten times more additions and multiplications than a second way. So if the second way takes a computer 15 minutes to do, the first way may take the computer, 150 minutes = 2[1/2] hours. Rules for arithmetic say how to change calculations without changing their results.

Changing the way a number or quantity is computed can also lead to formulas for other numbers or quantities. Examples of this have been given. Recall from the formula A = L·W for the area A of a rectangle, we obtained formulas for its length L and its width W. From the compound interest formula A = P(1+i)ⁿ we obtained formulas for P and i. (Aside: note the two roles of the letter A. It is an actor in both the area formula and in the compound interest formula. The letter A has one role or meaning when we look at the area formula and another role when we look at the compound interest formula.)
In this chapter, we will describe a large set of arithmetic rules and properties which say when two calculations or formulas give the same result. This knowledge allows us to replace one formula by another in a larger expression without changing whatever is being computed by the larger expression.

Postscript: Abuse of Equal Sign

The equal sign = is put between two symbols or expressions to say they have the same value.
(A) Writing 5(3 *2) = 6 = 30 is wrong since 6 does not have the same value as 30. But writing 5 (3 * 2) = 5 * 6 = 30 is right.
The solution of the equation

3 4

=

x 3

is given by x =3. But is an error, a mistake, a major misuse of the equal sign to insert an = 3 besides the x in the above equation to obtain 

3 4

=

x =3 3

in place of writing x = 3. While a person who writes 

x  = 3
3         

may mean x = 3, the expression 

x  = 3
 3         

actually means a third of x  is 3.  

Mathematics and English teachers should mark what is written, not was meant, so their students learn to write precisely. Precision is important. A person who does not write exactly what he or she means does not know how to read precisely what is written in their notes and textbooks,  and so is easily  confused.  Moreover, in mathematics, confusion about  notation, what is proper or not,  leads to errors in all calculations and in problem solving. Ouch!

2.1  Proper Use of the Equal Sign

The equal sign = can be used to say or suggest the following.

1. two different symbols (or expressions) are shorthand for the same number and quantity.
2. two different calculations or expressions give the same result when done, or
3. the value of a number or quantity can be computed using another expression.

The suggestion in question can be true or false depending on circumstances. Examples follow:

4+5	=	7+2
r2	=	r·r
3x+1	=	x+7
x+4	=	x+6

The first equation or equality holds (meaning is true) since both 4+5 and 7+2 are expressions giving the value 9.
The second equation r² = r·r always holds, no matter what value you give to r. It tells us how to compute the number or quantity described by the expression r².
The third equation 3x+1 = x+7 holds (is true) when and only when x = 2. When x has a value other than 2, the statement (suggestion or assertion) that 3x+1 gives the same result as x+7 is false.
The fourth statement x+4 = x+6 is always false. No value given to (or substituted for) x will make this statement true. Adding 4 and adding 6 to the same number give different results, no matter what the number is.

2.2  Replacement or Substitution

The box volume example met earlier gives a simple example in which replacement and substitution are used to tell us how to compute a quantity, the volume, in two different ways. Here is a reminder of the box volume calculation.

Flashback. Picture or imagine a box with a horizontal base. The box has a height H. The base of the box has an area A, a length L and a width W. In the formula V = H(WL) for the volume of the box, the parentheses tell us the calculation WL within them should be done first. The parentheses enclose or surround the subcalculation WL. The expression WL describes a calculation within another. It is a subformula. The symbol A and the subformula WL can replace each other. In the volume calculation, we can interchange them. They represent the same quantity, namely the base area of the box. From the calculation V = H(WL), by replacement of WL by its equal A we can find V = HA. To calculate the volume V using the latter calculation, we either use a known value of A or we find it from the formula A = W·L or from other information that might be available.

Algebra is based on obtaining and changing shorthand descriptions for the calculation of numbers and quantities, The replacement principle allows us to change how a number or quantity is computed. In changing and manipulating calculations, assume we can do the following.

1. Replace one expression by another when both give the same result (when computed).
2. Replace an expression by a single symbol, shorthand notation for the result of the expression.
3. Replace a symbol which is shorthand for the result of a calculation, by a calculation or expression which yields it. There might be several such calculations.
4. Replace a shorthand symbol (or expression) by the number or quantity its represents or gives when known, measured or computed.

These four abilities (rules) allows us to change and manipulate calculations, that is, go from one calculation to another, without changing the result that would be obtained. Algebraic shorthand expressions represent numbers, quantities and would-be calculations. The above four abilities give us the so-called replacement principle for the description of calculations: symbols or expressions can replace each other if they represent or result in equal numbers or quantities.

2.3  Real Numbers and Real Quantities

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:

1. the whole numbers 1, 2, 3, 4, 5, ....
2. the number zero: 0.
3. integers: 0, ±1, ±2, ±3, ±4, ±5, ±6, ....
4. 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.
5. 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. In higher-level mathematics, the convention is talk about real numbers instead of (signed) decimal numbers. If you are not familiar with signs, don't worry about them now. Worry later. 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.

2.4  Real Quantities

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.

Arithmetic Rules and Patterns

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

³ High school mathematics (circa 1990) talks only about real numbers, and leaves talk about quantities to physic courses and commerce courses. But calculations involve both real numbers and units of measurements. The convention in algebra textbooks is to emphasize the connection with real numbers but not real quantities. But in dealing with quantities in physical and monetary calculations, students need some guidance. Since the rules of algebra apply to calculations involving units, an algebraic tradition involving the manipulation of units and their powers needs to be presented and sanctioned in high school mathematics courses.

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.
Explaining Some Rules.
Next, you may meet more words than you ever wanted on the rules of arithmetic. Read on and look for the ideas new to you. Some, just a few, could be worth repeating to others.
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.

⁴You 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 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 more recent postscripts and site pages 3. Computation Rules and 4. Axioms & Computation Rules relate the above axioms or rules for algebra to the notion of equivalent computation rules. With the help of function notation for example, the associative law for addition is equivalent to saying that the two different computation rules

f(a,b,c) = a+(b+ c) and g(a,b,c) = (a+b)+c

will give the same result no matter what values are assigned to the variable or number place holders a, b and c. To learn and understand more read the more recent pages. That being said, this computation rule viewpoint was not given in my school days. In writing it, I think this viewpoint makes mathematical practices easier to understand and use. But I am wondering if this viewpoint departs from the spirit in which modern mathematics was written. Functions too were not regarded as equivalent computation rules in my school days.

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.

4  Products

4.1 Multiplication of Sums

The distributive property

(a+b) c = a c + b c

links addition and multiplication together. It says the calculation of (a+b) c and the calculation of a c+ b c both give the same result no matter what real numbers or real-valued expressions⁵

 ⁵expressions which give real numbers when computed

replace a, b and c. In manipulating a formula, we can replace either of these calculations by the other to get a new formula with the same result as the original one. An arithmetic example follows. Observe

(5+9)×10 = 14 ×10 = 140

and

5×10 +9×10 = 50+90 = 140

The replacement of (a+b) c by a c + b c is often called an expansion of (a+b) c. The reverse replacement, that of a c + b c by (a+b)c is called a factorization. The real number or expression playing the role of c is called the common factor. A common mistake in calculating (a+b)c is to forget the parentheses and calculate a+bc instead. The calculation of (5+9)×10 should give (14)×10 = 140 and not 5+9×10 = 5+90 = 95. The order in which arithmetic is done is important. Expressions within parentheses are calculated first.

Factorization Examples

1. 2x+4x = (2+4)x = 6x, with a common factor x,
2. 10y+17y = (10+17)y = 27y , with a common factor y,
3. 8  feet+16  feet = (8+16)  feet = 24 feet, with a common factor the foot. This unit should be spelt feet when more or less than one is expected. Or, the abbreviation ft can be used at all times.
4. 7  dollars+8  dollars = (7+8)  dollars = 15 dollars with common factor the dollar.
5. r²+3r² = (1+3)r² = 4r² with the common factor being r² = r·r,
6. 2r²+6r³ = (2+3r)2r² = 5r² with r³ = r·r² and the common factor again being r².

The other (left) distributive law c(a+b) = ca+cb can be used similarly.

The Non-Zero Product Rule from Decimal Arithmetic.

If you use decimal arithmetic to add two positive numbers together, the result will be positive. If you use decimal arithmetic to multiply two positive numbers together, the result will again be positive. This implies the Nonzero Product Rule indicated below in the setting of real numbers. .

We add another rule to those you have seen, namely if a and b are both nonzero real (signed decimal) numbers or real quantities then their product ab is also nonzero.⁶ This rule can be rewritten in contrapostive form: If a product ab of two real numbers a and b is zero then at least one of the factors a and b must be zero. The further explanation of this observation is an intellectual IOU. To collect, read about the contrapositive for one-way implication rules in the chapters on logic and reason below.

When a and b are real numbers. The contrapositive way of saying

If a and b are both nonzero then the product ab must be nonzero

is

If the product ab of two factors a and b is zero, then at least one of the factors a and b must be zero.

This zero product rule is used to obtain and justify the formula for the solution of the quadratic formula. If you have not met the quadratic formula, don't worry about it now. Later is sufficient. The presence of (supposedly) never disobeyed rules and properties in mathematics requires a knowledge of logic, that is rule- and pattern-based reason. In mathematics, the never-disobeyed rules are stated in terms of shorthand notation. So in mathematics after arithmetic, a knowledge of both algebraic shorthand notation and logic is needed.

Addition and Subtraction

Commutative Property

The commutative property for addition says

a+b = b+a

no matter what real numbers or real quantities replace a and b in the two calculations a+b and b+a.

  Example:   Replace a by 567.5 and b by 132.3. To calculate a+b = 567.5+132.3, we can add these two numbers as follows.

567.5

+132.3

______

To calculate b+a = 132.3+567.5, we can add these two numbers as follows.

132.3

+567.5

______

If you do the arithmetic carefully, you will see that the two results you get are both equal to 699.8 The order in which two (or more) numbers are added does not affect the answer.

On an electronic calculator, the addition a+b is done by entering the value 567.5 of the number a first and then the value 132.3 of the number b second. The addition of b+a is computed by entering these numbers in the other order, that is, 132.3 first and 567.5 second. Both calculations give the same result. So the order of addition is not important.

The commutative property of addition is not very interesting by itself, but we can use it with the replacement principle. In manipulating an equation, the commutative property for addition allows us to replace an expression of the form a+b in a formula by b+a, or vice-versa, as convenient. This replacement will give a new calculation with the same result (if we do the arithmetic it describes) as the original.

⁵expressions which give real numbers when computed
⁶Empirical Observation: If you use decimal arithmetic to add two positive numbers together, the result will be positive. If you use decimal arithmetic to multiply two positive numbers together, the result will again be positive. This implies the Nonzero Product Rule in the setting of decimal arithmetic.

ddition Associative Property

The associative property of addition is

(a + b) + c = a+ (b + c).

This associative property say the calculation of and expression of the form (a+b)+ c gives the same result as a+(b+ c). In formulas, either of these expressions can replace the other to get a new formula with the same result as the original one. Look at the following examples.

Here are three ways to add the numbers 42, 13 and 56 together. The different ways all give the same result. Check this.

1. Compute 42+13+56 all at once:

   42

   13

   ___+56

   ______

   The sum is left for you to compute.
2. Compute (42+13)+56 in the order indicated by the parentheses.
3. Compute 42+(13+56) in the order given by the parentheses.

For each order of addition, the common result is 111.

The (extended) associative property for additio

a+b+c = (a+b)+c = a+(b+c)

says three calculations, namely a+b+c, (a+ b)+ c, and a+ (b+ c) give the same result. In a formula, we can replace each of these by any of the others to get a new formula with the same result as the original one. The commutative and associative properties of addition allow us to change the order of additions and to group additions in anyway we please. The proof of this property is omitted. Because of this, parentheses to give or define an order of addition in sums are often omitted or dropped. Now when we describe calculations we may get a calculation of the form a+(b+c) in which the numbers a, b and c are given by expressions. In this case the computation of a+(b+c) still gives the same result as that of a+(b+c). So both a+(b+c) and (a+b)+c both give or represent the same result. So one can replace the other in any further calculation. In using the associative property

a+(b+c) = (a+b)+c = a+b+c,

we may let a, b and c be played by real numbers or real quantities, or by expressions which give real numbers or quantities.

Sums with the Number 0 - Adding or subtracting nothing

The expression a+0 represents a calculation in which nothing, namely zero 0 is added. The following equality

a+0 = a

gives an arithmetic property. In manipulating and changing formulas, this property may be used with 0 replaced by any expression which gives the result 0 when calculated. It is a useful trick. Watch for it.

Replacing Subtraction by Addition

The arithmetic properties described above involve only addition (and multiplication). But some of us know how to subtract. We can treat subtraction as an addition. After this is done, the above rules can be used. The opposite of addition is subtraction. Here (c+a)-a = c. Putting signs on numbers allows us to turn subtraction into addition. If you are not familiar with signed numbers, don't worry about the next few sentences: The negative of a number a is written (-a). The subtraction of a gives the same result as adding -a. That is, we rewrite an expression of the form b-a as b+(-a). When this is done, the properties of addition can be used with formulas involving subtraction.

Sum Grouping and Ordering

By applying chains of reason based on mathematical induction and starting with the associative and commutative properties for multiplication, it possible to show that the order in which a sum is computed should not affect the result. Here numbers in the sum can be added and grouped together in any order. This justifies sums being written without parentheses to indicate the order of addition in them.⁷

⁷A physical analogy for this is as follows: imagine umpteen bags of marbles, all of which are to be placed in a larger container. The total number of marbles put the larger container does not depend on the order in which the smaller bags are put in, and it does not depend on how the smallers bags are grouped together before they are put in. Discussing about this physical analogy departs from the pure development of mathematical concepts from the long chains of reasoning starting with rules or assumptions that involve no physics.

Products with the Number 1

Multiplication a number (or quantity) by 1 gives back that same number (or quantity respectively). That is,

a ×1 = a

whenever a represents a real number or quantity. Multiplying by 1 or an expression that has the value 1 is useful for changing units in the representation of a quantity. Advantage can be taken of the property that [(b)/(b)] = 1 when b ¹ 0 is a nonzero real number or quantity. The trick is based on the equalities a = a ·1 = a·[(b)/(b)] Example:   100 pennies = dollar suggests [100 pennies] / [1 dollar ] = 1. Now

2.50 dollar = 2.50 dollar ×1 = 2.50 dollar × 100 pennies 1 dollar = (2.50 ×100) pennies = 250 pennies

Multiplying by a fraction of the form [(b)/(b)] = 1 provides a way to change the way a calculation is done (or the units in it) without changing the result. The factor b here is to be chosen so that some convenient cancellation occurs. Note that other ways for changing the units with which a quantity is described are possible. We use each way as convenient, whichever appears to give the least amount of work or worry.

Product Grouping and Ordering

By applying chains of reason based on mathematical induction and starting with the associative and commutative properties for multiplication, it possible to show that the order in which a product is computed should not affect the result. Here terms in the product can be added and grouped together in any order. This justifies products (for instance terms in polynomials) being written without parentheses to indicate the order of multiplication in them.

7  Powers: Bases and Exponents

Instead of writing r ×r, we write r². Instead of writing r ·r ·r, we write r³. Let n be a whole number. The expression rⁿ stands for r times itself n times. For example,

• r¹ = r,
• r² = r·r,
• r³ = r²·r = (r·r)·r,
• r⁴ = r³·r = ((r·r)·r)·r,
• r⁵ = r⁴·r = (((r·r)·r)·r)·r ,
• r⁶ = r⁵·r = ¼

(The pattern here is that if you know how to compute rⁿ, you can compute rⁿ⁺¹ by computing rⁿ ·r.) For notational convenience, we put r⁰ = 1. We also put r^-n = [1/(rⁿ)] with the proviso (condition) that division by zero is not allowed. The letter r above can be replaced by any other symbol or shorthand expression for a real number or quantity. Some examples with numbers follow.

22 = 2·2 = 4 103 = (102)·10 = 100 ·10 = 1000 54 = 53 ·5 = 5·5 ·5 ·5 = 25 ·25 = 625

The following equalities hold when you deal with powers. Here let n and m represent whole numbers.

rn·rm = r(n+m) rn rm = r(n-m) = 1 rm-n .

More can be said, but that is left for another text.

8  Division - Its Replacement by a Multiplication

The arithmetic properties described above involve only addition (and multiplication). But some of us know how to divide. We can treat division as a multiplication. After this is done, the above rules involving multiplication and addition can be used when division is present. Before describing how this is done, we note:

1. (c ×a)¸a = c.
2. Each fraction [(a)/(b)] = a ×b.
3. The reciprocal of a nonzero number or quantity b is the fraction p = [1/(b)] = 1¸b. It is the only number (or quantity) p with the property that p b = 1.
4. Division by the real number zero is not possible. So whenever we try to divide we must add the condition that the divisor b is nonzero - In evaluating the ratio of two expressions, the divisor after all our efforts might equal zero. If a divisor happens to equal zero, our calculation must stop. Division by zero is not defined. Because of this, a calculator will give you an error message when you try [accidentally I presume] to divide by zero.

The key idea for the replacement of a division by a multiplication with the same effect is as follows. Division by a number b gives the same result as multiplying by its reciprocal 1¸b =[1/b].     That is,

a b = a¸b = a · 1 b

whenever the divisor b is nonzero and a is a real number or quantity. This is a property of arithmetic: another rule to remember as needed. Since division by a nonzero, real number b has the same effect as multiplying by its reciprocal 1¸b =[1/b], we can rewrite all divisions as multiplications. With this replacement, the multiplication properties of arithmetic can be used to replace a formula involving division or fractions by another involving multiplication.

Rules for Division or Fractions

where denominators and numerators are whole numbers, integers or real numbers, complex numbers or polynomials etc.

By replacing subtractions and divisions in formulas with additions and multiplications, we get formulas only involving additions and multiplication. These new formulas give the same result as the original ones. They can be changed, say rephrased, using the properties of real numbers and quantities given above. After this, for cosmetic reasons depending on circumstances, some multiplications and additions might be replaced by divisions and subtractions.
The rules for doing arithmetic with fractions and divisions can be obtained from the properties of real numbers if we use the equality

a b

= a

.

1 b

 to define the "fraction" a/b whenever (a,b) is a pair of real numbers with the second number b non-zero.
So instead of properties of addition and multiplication, you can use the following rules which say when two different fractional expressions give the same results.  These rules provide methods for arithmetic operations on fractions where the numerators and denominators are real (or complex) numbers or polynomial expressions whose values are real (respectively, complex) numbers.
First, the cancellation rule says

a c b c = a b

whenever a, b and c are both nonzero real numbers or quantities. Here there is no condition on a other than it be a real number or quantity as well. (Remember division by zero is not permitted as division by zero is not defined.) Next, one fraction addition method gives

a b + c d = ad+cb bd

whenever b and d denote nonzero real numbers.

But if we use a common denominator M, we can rewrite the foregoing as

a b + c d = a(M×c)+c(M×d) M

In the case where numerators and denominators are given by integers or polynomial expression, simplification of the expression of the right requires less work if M is taken to be the least common multiple of the denominators b and d.
Finally, we state the nameless rule

a· b c = ab c

for multiplying a fraction by a real number a, or a real-valued expression a.

9  Subtraction Revisited

Recall subtraction of a number or quantity a gives the same result as adding its negative -a - also called its additive inverse. For example, we may read 8-5-2 as 8+(-5)+(-2). By the associative and commutative properties of addition described below the order and grouping of addition (unlike subtraction or division) is not important. You can sum 8+(-5)+(-2) in any order you please. The result is still 1. In general,

a-b-c = a+(-b)+(-c)

Now the way decimal arithmetic is done and written today is a convention which began roughly four hundred years ago. This convention is still young. Mathematicians have not so far defined the meaning of expressions like 5 ×6¸2. This situation is about to be changed. The treatment of division can be made simpler:
In analogy with the traditional definition and interpretation of the expression 5-6-2 as 5+(-6)+(-2), we may take 5 ÷6 ÷ 2 to be 5 ·[1/6]·[1/2] since division by a number gives the same result as multiplying by its reciprocal. More generally, we take

a÷b ÷ c = a 1 b 1 c ·

Then a ÷b ÷c = a ÷(bc) (why?) With this convention, the treatment of multiplication and divisions becomes similar to that of addition and subtractions. This proposal gives a departure from the convention that a expression like a÷b ÷c should be left undefined because the order of division is not clear. Here is one more convention that can be safely adopted. Just as -a = (-1)a, we could let ÷a = [1/(a)] provided a is nonzero.