ALGEBRA is a method of written calculations. And a calculation is replacing one set of symbols with another. In arithmetic we may replace the symbols '2 + 2' with the symbol '4.' In algebra we may replace 'a + (−a)' with '0.'
a + (−a) = 0.
A formal rule, then, shows how an expression written in one formmay be rewritten in 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 statement p with statement q. For example,
x + a = b implies x = b − a.
That 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 |
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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) |
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
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 | = Any number. |
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 |
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(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 |
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| 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
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
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)
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 | = | anbn | | Power of a product of factors |
|
|
|
(am)n | = | amn | | Power of a power |
24. The definition of a negative exponent
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
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 xn = n log x.
32. The definition of the complex unit i
i 2 = −1