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The Formal Rules of Algebra

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 form may 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
 
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 not equal 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

ab = 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  ba  to  ab

ba = −(ab)

Now,  b + a  is equal to a + b.  But  ba  is the negative of ab.

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 not equal 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
 
(xa)(xb) = x2 − (a + b)x + ab  
  Quadratic trinomial
 
(a ± b)2 = a2 ± 2ab + b2 Perfect square trinomial
 
(a + b)(ab) = a2b2 The difference of
  two squares
 
(a ± b)(a2 algebra 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,         xa  =  b,
 
then     then  
  x  =  ba.     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)

algebra

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
 
algebra
 
(am)n  =  amn   Power of a power

24.  The definition of a negative exponent

an  =   1 
an

25.  The definition of exponent 0

a0 = 1

26.  The definition of the square root radical

algebra

The square root radical squared produces the radicand.

27.  Equations of the form  a2 = b

If
a2  =  b,
 
then
a  =  ±algebra.

28.  Multiplying/Factoring radicals

algebra  =  algebra
 
and symmetrically,
 
algebra  =  algebra

29.  The definition of the nth root

algebra

30.  The definition of a rational exponent

algebra

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

End of the Lessson

Next Topic:  Rational and irrational numbers


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