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6 ## RECIPROCALSAND ## ZEROThe quotient of TWO NUMBERS ARE CALLED reciprocals of one another if their product is 1.
The student should not do arithmetic -- that is, "cancel" the
Problem 1. Write the symbol for the reciprocal of 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 necessary to know what "log 2" means The statement
is purely
To "prove" anything, we must satisfy its definition. In this case, the definition of
Problem 5.
1 over any number means: The reciprocal of that number.
Problem 6.
The definition of division We say in algebra that division is
"a divided by b is equal to a times the Equivalently, We may rewrite any fraction as the numerator That rule in the box is called The Definition of Division. Division is defined in terms of multiplication, just as subtraction can be defined in terms of addition: In a sense, we do not need the word "division." We could give the purely formal definition:
Problem 7. Rewrite each of the following according to The Definition of Division.
To divide by ¼ is to multiply by its reciprocal, 4.
In other words, when the numerator and denominator are equal, you should see immediately that the fraction is equal to 1. This has nothing to do with "canceling." It is a Problem 9. Evaluate each of the following. (Assume that no denominator is 0.)
Inverse operations Division and multiplication are inverse operations. That means that if you start with some number
Or, if you first multiply by
Problem 10. Complete each equality with some operation, so that the equality is preserved.
The student may think that is trigonometry, but it is not. It is algebra.
The rule called the Definition of Division --
-- is is merely a formal rule. It tells us that we may replace with 12 . But it does not tell us how to evaluate that division. For that, we must return to arithmetic and to the relationship between division and multiplication. 12 ÷ 3 = ? What number Equivalently, how many times do we have to Thus, if the quotient of
-- then
For example,
because 4 We will be applying this below. Rules for 0
"If any factor is 0, the product will be 0." Problem 11.
Problem 12. If the product of two factors is 0,
what can you conclude about
Either
Problem 13. Which values of ( The product will equal 0 only when one of the three factors equals 0.
The first factor will equal 0 when
The middle factor when
The last factor when
For those three values of We will now investigate the following forms. In each one,
See See above.
There is (The student should not confuse
We say that it is In summary (
(For a "proof" that 1 + 1 = 1, click here.) One sometimes hears that a number divided by 0 equals infinity; that is, an infinite number of 0's will equal that number. But that would imply that an infinite number of 0's will equal every number, from which it follows that all numbers are equal. It is also common to hear that 8/0 is "undefined." Rather, it is
is undefined for For we can define 0/0 to be any number. However, we can never define the symbol 8/0. Problem 15.
Problem 16. Let
Problem 17. Does 0 have a reciprocal? No. There is no number, according to the definition of a reciprocal, that when multiplied with 0 will produce 1.
Problem 18. What is the only way that a fraction could be equal to 0? The numerator must be equal to 0.
Problem 19. Solve for
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