The Formal Rules of Algebra
The student who has had a course in algebra, will find this a complete review. The student who is now taking such a course, will see what is in store.
ALGEBRA is a method of written calculations. And what is a calculation but replacing one set of symbols with another? In arithmetic we replace '2 + 2' with '4.' In algebra we may replace 'a + (−b)' with 'a − b.'
a + (−a) = a − b.
We call that a formal rule. It shows how an expression written in one form may be replaced with 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,
p implies q,
means: We may replace—in the sense of follow—statement p with statement q. For example,
x + a = b implies x = b − a.
That means that we may follow the statement 'x + a = b' with the statement 'x = b − a.'
For we solve equations by a logical sequence of statements.
Algebra depends on how things look. We can say, then, that algebra is a system of formal—grammatical—rules. What follows are what we are permitted to write.
(See the complete course, Skill in Algebra.)
11. The axioms of "equals"
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
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,
14. The additive inverse of a: −a
a + (−a) = −a + a = 0
The "inverse" of a number undoes what the number does.
Two numbers are called reciprocals of one another if their product is 1.
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.
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
a(−b) = −ab. (−a)b = −ab. (−a)(−b) = ab.
"Like signs produce a positive number; unlike signs, a negative number."
11. Rules for 0
a· 0 = 0· a = 0.
If a 0, then
Division by 0 is an excluded operation. (Skill in Algebra, Lesson 5.)
13. The same operation on both sides of an equation
We may add the same number to both sides of an equation;
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:
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
17. Change of sense when solving an inequality
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
We may multiply both the numerator and denominator by the same factor; we may divide both by a common factor.
20. Multiplication of fractions
21. Division of fractions (Complex fractions)
22. Addition of fractions
The common denominator is the LCM of denominators.
23. The rules of exponents
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
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
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