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.
1. 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.
2. The commutative rules of addition and multiplication
a + b = b + a
a· b = b· a
3. The identity elements of addition and multiplication:
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.
4. 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.
5. The multiplicative inverse or reciprocal of a, 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.
6. The algebraic definition of subtraction
a − b = a + (−b)
Subtraction, in algebra, is defined as addition of the inverse.
7. 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.
8. The inverse of the inverse
−(−a) = a
9. 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 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 is not 0 then
Then
0/a = 0
a/0 is no value or we can not defined
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 a = b then a + c = b + c.
if a = b then 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.
15. Change of sign on both sides of an inequality:
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
Four Operations and their inverses
if x + a = b then x = b - a
if x - a = b then x = b + a
if x.a = b then x = a/b
if x/a = b then x = a.b
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 = (a.x)/(a.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) = (a.c)/(b.d)
a.(b/c) = (a.b)/c
21. Division of fractions (Complex fractions)
(a/b)/(c/d) = (a/b).(d/c) = (a.d)/(b.c)