6 ## SOME RULESOF ## ALGEBRAALGEBRA, we can say, is a body of formal rules. They are rules that show how something written one form may be rewritten in another form. For what is a calculation if not transforming one set of symbols into another?
In arithmetic we may replace the symbols '2 + 2' with this symbol '4.' In algebra, we may replace ' Here are some of the basic rules of algebra:
Associated with these -- and with any rule -- is the rule of symmetry: If For one thing, this means that a rule of algebra goes both ways. Since we may write
We may replace The rule of symmetry also means that
And so the rules of algebra tell us what we are allowed to write. They tell us what is legal. Problem 1. Use the rule of symmetry to rewrite each of the following. And note that the symmetric version is also a rule of algebra. To see the answer, pass your mouse over the colored area.
The order of terms does not matter. We express this in algebra by writing
That is called the commutative rule of addition. It will apply to any number of terms.
The order does not matter.
Example 1. Apply the commutative rule to
* Here is the commutative rule of multiplication:
The order of factors does not matter.
The rule applies to any number of factors. What is more, we may associate factors in any way: ( And so on.
Example 2. Multiply 2
2 It is the style in algebra to write the numerical factor to the left of the literal factors. Problem 2. Multiply.
Problem 3. Rewrite each expression by applying a commutative rule.
We have seen the following rule for 0 (Lesson 3 ): For any number
0 added to any number does not change the number. 0 is therefore called the identity of addition. The inverse of adding The inverse of an operation undoes that operation. If we start with 5, for example, and then add 4, 5 + 4, then to undo that -- to get back to 5 -- we must add −4: 5 + 4 + (−4) = 5 + 0 = 5. Adding −4 is the inverse of adding 4, and vice-versa. We say that −4 is the additive inverse of 4. In general, corresponding to every number
A number combined with its inverse gives the identity. We have seen that that rule is essentially the definition of − Thus, the additive inverse of −(− Problem 4. Transform each of the following according to a rule of algebra.
g) sin The student might think that this is trigonometry, but it is not. It is Problem 5 . Complete the following.
g) tan Two rules for equations An equation is a statement that two things -- the two sides -- are equal. Inherent in the meaning of
The rule means: We may This is the algebraic version of the axiom of arithmetic and geometry:
-- upon adding 2 to both sides.
-- upon subtracting 2 from both sides. But the rule is stated in terms of addition. Why may we subtract? Because subtraction is equivalent to addition of the negative.
Therefore, any rule for addition is also a rule for subtraction.
In Example 4, the effect of subtracting 2 from both sides is to transpose +2 to the other side of the equation as −2. We will go into this more in Lesson 9. Problem 6.
This rule means: We may Example 5. If
Now, what happened to 2 We multiplied it by 5. Therefore, to preserve the equality, we must multiply 3 by 5, also. 10 Example 6. If
Here, we multiplied both sides by 2, and the 2's simply cancel. See Lesson 26 of Arithmetic, Example 5. Example 7. If
Here, we Because division is equal to multiplication by the reciprocal. In this example, we could say that we multiplied both sides by ½. Any rule for multiplication, then, is also a rule for division. Problem 7.
Problem 9. Changing signs on both sides. Write the line that results from multiplying both sides by −1.
This problem illustrates the following theorem: In any equation we may change the
This follows directly from the uniqueness of the additive inverse.
Which is what we wanted to prove. We will have occasion to apply this theorem when we come to solve equations. For we will see that to "solve" an equation we must isolate Problem 10.
Thus if (If Next Lesson: Removing grouping symbols Please make a donation to keep TheMathPage online. Copyright © 2015 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |