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, thenq,
or equivalently
pimpliesq,
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 / IdentityIf 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 + aa·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.
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 is1.
Thus, 1/a symbolizes that number which, when multiplied by a, produces 1.
q / is / q
p / .
1
6. 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
ab / = / 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
0a / =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 / Ifa / = / 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
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
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.
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
xy / = / 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
ab / · / 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
ac / + / 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 dividingam
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 / = / 1an
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
Ifa² / = / 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 / xy / =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