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The Formal Rules of Algebra

The student who has had a course in algebra, will find this a complete review. The student who is now taking such a course, will see what is in store.

ALGEBRA is a method of written calculations. And what is a calculation but replacing one set of symbols with another? In arithmetic we replace '2 + 2' with '4.' In algebra we may replace 'a + (−b)' with 'a − b.'

a + (−b) = a − b.

We call that a formal rule. It shows how an expression written in one form may be replaced with a different form. The = sign means "may be rewritten as" or "may be replaced by."

If p and q are statements (equations), then a rule

If p, then q,

or equivalently

p implies q,

means: We may replace—in the sense of follow—statement p with statement q. For example,

x + a = b implies x = b − a.

That means that we may follow the statement 'x + a = b' with the statement 'x = b − a.'

For we solve equations by a logical sequence of statements.

Algebra depends on how things look. We can say, then, that algebra is a system of formal—grammatical—rules. What follows are what we are permitted to write.

(See the complete course, Skill in Algebra.)

11. The axioms of "equals"

a = a		Identity

If a = b, then b = a.		Symmetry

If a = b and b = c, then a = c.		Transitivity

It is not possible to give an explicit definition of the word "equals," or its symbol = . Those rules however are an implicit definition. The meaning of "equals" implies those three rules.

As for how the rule of symmetry comes up in practice, see Lesson 6 of Algebra. The rule of symmetry applies to all of the rules below.

12. The commutative rules of addition and multiplication

a + b	=	b + a

a· b	=	b· a

13. The identity elements of addition and multiplication:

3. 0 and 1

a + 0 = 0 + a = a

a· 1 = 1· a = a

Thus, if we "operate" on a number with an identity element,
it returns that number unchanged.

14. The additive inverse of a: −a

a + (−a) = −a + a = 0

The "inverse" of a number undoes what the number does.
For example, if you start with 5 and add 2, then to get back to 5 you must add −2. Adding 2 + (−2) is then the same as adding 0 -- which is the identity.

15. The multiplicative inverse or reciprocal of a,
5. symbolized as	1 a	(a 0)

a·

1
a

· a

= 1.

Two numbers are called reciprocals of one another if their product is 1.
Thus, 1/a symbolizes that number which, when multiplied by a, produces 1.

The reciprocal of

p
q

q
p

16. The algebraic definition of subtraction

a − b = a + (−b)

Subtraction, in algebra, is defined as addition of the inverse.

17. The algebraic definition of division

a
b

a·

1
b

Division, in algebra, is defined as multiplication by the reciprocal.
Hence, algebra has two fundamental operations: addition and multiplication.

18. The inverse of the inverse

−(−a) = a

19. The relationship of b − a to a − b

b − a = −(a − b)

Now, b + a is equal to a + b. But b − a is the negative of a − b.

10. The Rule of Signs for multiplication, division, and
10. fractions

a(−b) = −ab. (−a)b = −ab. (−a)(−b) = ab.

a
−b

= −

a
b

−a
b

= −

a
b

−a
−b

a
b

"Like signs produce a positive number; unlike signs, a negative number."

11. Rules for 0

a· 0 = 0· a = 0.

If a 0, then

0
a

= 0.

a
0

= No value.

0
0

= Indeterminate.

Division by 0 is an excluded operation. (Skill in Algebra, Lesson 5.)

12. Multiplying/Factoring

m(a + b) = ma + mb	The distributive rule/
	Common factor

(x − a)(x − b) = x² − (a + b)x + ab
	Quadratic trinomial

(a ± b)² = a² ± 2ab + b²	Perfect square trinomial

(a + b)(a − b) = a² − b²	The difference of
	two squares

(a ± b)(a² ab + b²) = a³ ± b³	The sum or difference of
	two cubes

13. The same operation on both sides of an equation

If					If

	a	=	b,			a	=	b,

then				then

a + c		=	b + c.			ac	=	bc.

We may add the same number to both sides of an equation;
we may multiply both sides by the same number.

14. Change of sign on both sides of an equation

If

	−a	=	b,

then

	a	=	−b.

We may change every sign on both sides of an equation.

15. Change of sign on both sides of an inequality:
15. Change of sense

If

	a	<	b,

then

	−a	>	−b.

When we change the signs on both sides of an inequality, we must change the sense of the inequality.

16. The Four Forms of Equations corresponding to the
16. Four Operations and their inverses

If				If

	x + a	=	b,		x − a	=	b,

then				then
	x	=	b − a.		x	=	a + b.

If					If

	ax	=	b,			x a	=	b,

then					then
	x	=	b a	.		x	=	ab.

See Skill in Algebra, Lesson 9.

17. Change of sense when solving an inequality

If

	−ax	< b,

then

	x	> −	b a	.

18. Absolute value

If |x| = b, then x = b or x = −b.

If |x| < b then −b < x < b.

If |x| > b (and b > 0), then x > b or x < −b.

19. The principle of equivalent fractions

x y	=	ax ay

and symmetrically,
ax ay	=	x y

We may multiply both the numerator and denominator by the same factor; we may divide both by a common factor.

20. Multiplication of fractions

a b	·	c d	=	ac bd

a ·		c d	=	ac d

21. Division of fractions (Complex fractions)

algebra

Division is multiplication by the reciprocal.

22. Addition of fractions

a c	+	b c	=	a + b c	Same denominator

a b	+	c d	=	ad + bc bd	Different denominators with no common factors

a bc	+	e cd	=	ad + be bcd	Different denominators with common factors