Summary of algebra - Topics in precalculus

Topics in

P R E C A L C U L U S

The Formal Rules of Algebra

ALGEBRA is a method of written calculations. A formal rule shows how an expression written in one form may be rewritten in a different form. For example,

a + b = b + a.

This means that if we see something that looks like this

a + b

then we are allowed to rewrite it so that it looks like this

b + a.

In a formal rule, the = sign means "may be rewritten as" or "may be replaced by." For, what is a calculation but replacing one set of symbols with another? In arithmetic, we may replace '2 + 2' with '4.' In algebra, we may replace 'a + b' with 'b + a.'

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

If p, then q,

p implies q,

means: We may replace statement p with statement q. For example,

x + a = b implies x = b − a.

This means that we may replace the statement 'x + a = b' with the statement 'x = b − a.'

Algebra depends on how things look. We can say, then, that algebra is a system of formal rules. The following 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

These are the "rules" that govern the use of the = sign.

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)

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

"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

= Any number.

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, then x > b or x < −b.

19. The principle of equivalent fractions

x y	=	ax ay

and symmetrically,
ax ay	=	x y

Both the numerator and denominator may be multiplied by the same factor; both may be divided by the same factor.

20. Multiplication of fractions

a b	·	c d	=	ac bd

a ·		c d	=	ac d

21. Division of fractions (Complex fractions)

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

The common denominator is the LCM of denominators.

23. The rules of exponents

a^maⁿ	=	a^m+n	Multiplying or dividing

a^m aⁿ	=	a^m−n	powers of the same base

(ab)ⁿ	=	a1ⁿbⁿ	Power of a product of factors



(a^m)ⁿ	=	a^mn	Power of a power