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 equivalently
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
1
= ·a = 1
a
a
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
1
p
q
q
is p .
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
a
1
= a·
b
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
a
−a
a
−a a
=− .
=− .
= .
−b
b
b
b
−b b
"Like signs produce a positive number; unlike
signs, a negative number."
11. Rules for 0
a· 0 = 0· a = 0
If a
0
= 0.
a
0, then
a
= No value.
0
0
= Any number.
0
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,
then
a = b,
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
x
= b,
a
ax = b,
then
then
b
a
x= .
x = ab.
17. Change of sense when solving an
inequality
If
−ax < b,
then
b
a
x>− .
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 ax
=
y ay
and symmetrically,
ax x
=
ay 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 c ac
· =
b d
bd
a·
c ac
=
d
d
21. Division of fractions (Complex
fractions)
Division is multiplication by the reciprocal.
22. Addition of fractions
a b
a+b
+ =
Same denominator
c c
c
a
c
ad + bc
+ =
b d
bd
Different denominators with
no common factors
a
e
ad + be
+ =
bc
cd
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
= am−n
an
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
1
a−n = 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
= log x − log y.
y
log xn = n
log x.
log 1 = 0.
logbb = 1.
32. The definition of the complex
unit i
i ² = −1