1
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,
or
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 |
= | 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 |
is | 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 |
= − | 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
| aman | = | am+n | Multiplying or dividing | |
| am an |
= | am−n | powers of the same base |
|
| (ab)n | = | a1nbn | Power of a product of factors | |
 |
||||
| (am)n | = | amn | Power of a power | |
24. The definition of a negative exponent
| a−n | = | 1 an |
25. The definition of exponent 0
a0 = 1
26. The definition of the square root radical
The square root radical squared produces the radicand.
27. Equations of the form a² = b
| If | |||
| a² | = | b, | |
| then | |||
| a | = | ± . |
|
28. Multiplying/Factoring radicals
 |
= |  |
|
| and symmetrically, | |||
|
= | |
|
29. The definition of the nth root
30. The definition of a rational exponent
It is more skillfull to take the root first.
31. The laws of logarithms
log xy = log x + log y.
| log | x y |
= log x − log y. |
log xn = n log x.
| log 1 = 0. | logbb = 1. |
32. The definition of the complex unit i
i ² = −1
Next Topic: Rational and irrational numbers
www.proyectosalonhogar.com