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6 RECIPROCALS AND ZERO The quotient of a divided by b TWO NUMBERS ARE CALLED reciprocals of one another when their product is 1.
The student should not do arithmetic -- that is, "cancel" the a's. The
multiplied with a it produces 1, which is the identity Problem 1. Write the symbol for the reciprocal of z. To see the answer, pass your mouse over the colored area.
It is that number which, when multiplied with log 2, produces 1. And it is not even necessary to know what "log 2" means! The statement
is purely formal.
To "prove" anything, we must satisfy its definition. In this case, the definition of reciprocals is that their product is 1.
Problem 5.
Problem 6.
The definition of division We say, in algebra, that division is multiplication by the reciprocal.
a divided by b is equal to a times the reciprocal of b. Equivalently, We may rewrite any fraction as the numerator That rule in the box is called The Definition of Division. Division is stated in terms of multiplication, just as subtraction can be stated in terms of addition: a − b = a + (−b). (Lesson 3.) Problem 7. Rewrite each of the following according to The Definition of Division.
In other words, when the numerator and denominator are equal, the fraction is immediately equal to 1. This has nothing to do with "canceling." Problem 9. Evaluate each of the following. (Assume that no denominator is 0.)
Now the rule called the Definition of Division --
But it does not tell us how to evaluate it. For that, we must return to arithmetic, and to the relationship between division and multiplication. The quotient of a divided by b,
is that number q such that q times b is equal to a. a = qb We will be applying this below. Rules for 0 a· 0 = 0· a = 0 "If any factor is 0, the product will be 0." Problem 10.
Problem 11. If the product of two factors is 0, ab = 0, what can you conclude about a or b? Either a = 0 or b = 0. Problem 12. Which values of x will make this product equal 0? (x − 1)(x + 2)(x + 3) = 0 Each value of x that will make each factor equal to 0. In the first factor, x = 1. In the middle factor, x = −2. And in the last factor, x = −3. For those three values -- and only those three -- will that product equal 0.
We will now investigate the following forms. In each one, a0.
Problem 13.
n = 4. Because according to the meaning of the quotient n, 4· 2 = 8.
n = 0. Because 0· 8 = 0.
There is no number n such that n· 0 = 8. Division by
meaningless symbol. It has no value. (The student should not confuse no value with the value 0. 0 is a perfectly good number.)
n could be any number, because any number n· 0 = 0. The
In summary (a0):
(For a "proof" that 1 + 1 = 1, click here.)
is "undefined" for x = 0. Nevertheless, since
for all other values of x, we can define that function as 5 -- or any number we please -- when x = 0. And
Problem 14.
Problem 15. Let x = 2, and evaluate the following.
Problem 16. Does 0 have a reciprocal?
Problem 17. What is the only way that a fraction could be equal to 0? The numerator must be equal to 0. Problem 18. Solve for x.
See the previous problem. Next Lesson: Removing grouping symbols Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |