6
RECIPROCALS
AND
ZERO
The quotient of a divided by b
TWO NUMBERS ARE CALLED reciprocals of one another if their product is 1.
| The reciprocal of 2 is | 1 2 |
-- because |
| 2 · | 1 2 |
= 1. |
| In general, the symbol for the reciprocal of any number a is | 1 a |
. |
| a · | 1 a |
= | 1 a |
· a | = 1 |
The student should not do arithmetic -- that is, "cancel" the a's. The
| student should recognize by the written form itself that | a · | 1 a |
= 1. |
Problem 1. Write the symbol for the reciprocal of z.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
|
1 z |
| Problem 2. If pq = 1, then q = | 1 p |
. q is the reciprocal of p. |
| Problem 3. What is the meaning of the symbol | 1 log 2 |
? |
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
| 1 log 2 |
· log 2 = 1 |
is purely formal. It depends only on how it looks.
| Problem 4. Prove that the reciprocal of | 3 4 |
is | 4 3 |
. |
| 3 4 |
· | 4 3 |
= 1. |
To "prove" anything, we must satisfy its definition. In this case, the definition of reciprocals is that their product is 1.
| In general, the reciprocal of a fraction | a b |
is | b a |
. |
Problem 5.
| a) |  |
= | 2. |
1 over any number means: The reciprocal of that number.
| b) |  |
= | 3 2 |
Problem 6.
| a) | xyz · | 1 xyz |
= | 1 | b) | (a − b) | · | 1 a − b |
= 1 |
| c) | (p + q) · | 1 (p + q) |
= 1 | d) | &8% · | 1 &8% |
= | 1 |
The definition of division
We say in algebra that division is multiplication by the reciprocal.
| a b |
= a · | 1 b |
= | 1 b |
· a |
"a divided by b is equal to a times the reciprocal of b."
Equivalently,
We may rewrite any fraction as the numerator
times the reciprocal of the denominator.
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: a − b = a + (−b). (Lesson 3.)
In a sense, we do not need the word "division." We could give the purely formal definition:
| This symbol | a b |
means | a · | 1 b |
Problem 7. Rewrite each of the following according to The Definition of Division.
| a) | 3 4 |
= | 3 · | 1 4 |
b) | x 2 |
= | x · | 1 2 |
or ½x |
| c) | x + 1 x + 2 |
= | (x + 1) · | 1 x + 2 |
| d) | a + b + c 6 |
= | (a + b + c) · | 1 6 |
| e) | x ¼ab |
= | 4x ab |
To divide by ¼ is to multiply by its reciprocal, 4.
This shows how not to leave a fraction in the denominator.
| f) |  |
= | 3a 2b |
| Problem 8. Prove that | a a |
= 1 , for any number a 0. |
| a a |
= | a · | 1 a |
, according to the Definition of Division, |
| = | 1, | according to the definition of reciprocals. | ||
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 formal rule.
Problem 9. Evaluate each of the following. (Assume that no denominator is 0.)
| a) | x − 2 x − 2 |
= | 1 | b) | a + b + c a + b + c |
= | 1 | c) | −(x + 5) −(x + 5) |
= | 1 |
| d) | x −x |
= | −1 | e) | a + b −(a + b) |
= | −1 | f) | −(x²
+ 5x − 2) x² + 5x − 2 |
= | −1 |
Inverse operations
Division and multiplication are inverse operations. That means that if you start with some number x and then divide it by a number a, then if you want to return to x, you have to multiply by a.
| x a |
· | a | = | x. |
Or, if you first multiply by a, then to undo that you have to divide by a.
| x · a · | 1 a |
= | x |
| Or, equivalently, multiply by | 1 a |
. |
| 1 a |
is also called the multiplicative inverse of a, |
| because when we multiply it with a, it produces 1, which is the identity. | |
| Compare Lesson 11 of Arithmetic: Property 1 of division. | |
Problem 10. Complete each equality with some operation, so that the equality is preserved.
| a) 27 = 27 · 3 | · | 1 3 |
| b) 27 = | 27 3 |
· | 3 |
| c) p = | p q |
· | q |
| d) m = mn | · | 1 n |
| e) cos x = | cos x sin x |
· | sin x |
The student may think that is trigonometry, but it is not. It is algebra.
| f) cos x = 2 cos x sin x | · |
__1__ 2 sin x |
| The quotient of | a b |
The rule called the Definition of Division --
| a b |
= a · | 1 b |
-- 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 times the divisor will equal the dividend?
Equivalently, how many times do we have to add the divisor so that it will equal the dividend?
Thus, if the quotient of a divided by b is some number n --
| a b |
= | n |
-- then n must be that number such that n times b is equal to a.
nb = a.
For example,
| 12 3 |
= | 4, |
because
4 · 3 = 12.
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 11.
| a) | 9 · 0 = 0 | b) | 0 · 9 = 0 | c) | 7· 45 · 127 · 0 · 39 = 0 |
Problem 12. 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 13. Which values of x will make this product equal 0?
(x − 1)(x + 2)(x + 3) = 0
The product will equal 0 only when one of the three factors equals 0.
The first factor will equal 0 when x = 1. (Lesson 2.)
The middle factor when x = −2.
The last factor when x = −3.
For those three values of x -- and only those three -- will that product equal 0.
We will now investigate the following forms. In each one, a
0.
| 0 a |
= ? |
| a 0 |
= ? |
| 0 0 |
= ? |
| a) | 8 2 |
= n. What number is n? |
n = 4. Because according to the meaning of the quotient n, 4 · 2 = 8.
See See above.
| b) | 0 8 |
= n. What number is n? |
n = 0. Because 0 · 8 = 0.
| Therefore, | 0 a |
= 0, for any number a 0. |
| c) | 8 0 |
= n. What number is n? |
There is no number n such that n · 0 = 8. Division by 0 is an excluded operation. The symbol 8/0, although it may look like a number, it is not a number. 8/0 is a meaningless symbol. It has no value.
(The student should not confuse no value with the value 0. 0 is a perfectly good number.)
| d) | 0 0 |
= n. What number is n? |
n could be any number, because for any number n, n · 0 = 0.
| The symbol | 0 0 |
could be defined to have any value. |
We say that it is indeterminate.
In summary (a0):
| 0 a |
= 0. |
| a 0 |
= No number. |
| 0 0 |
= Indeterminate. |
(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 undefinable. Elsewhere in mathematics, for example, we say that this quotient
| x2 − 4 x − 2 |
is undefined for x = 2. Nevertheless, by defining 0/0 to be 1, it is possible to define that quotient when x = 2.
For we can define 0/0 to be any number.
However, we can never define the symbol 8/0.
Problem 15.
| a) | 0 5 |
= | 0. | b) | 0 −5 |
= | 0. | c) | 0 x |
(x0) = |
0. |
| d) | 5 0 |
= | No number. | e) | 0 0 |
= | Any number. | f) | x 0 |
= | No number. |
Problem 16. Let x = 2, and evaluate the following.
| a) | x − 2 x + 2 |
= | 0 4 |
= 0. | b) | x + 2 x − 2 |
= | 4 0 |
= No value. |
| c) | 2x − 4 3x − 6 |
= | 0 0 |
= Any value. |
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.
| Equivalently, there is no number | 1 0 |
. |
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 x.
| a) | x − 2 x + 5 |
= 0. | b) | x + 3 x − 1 |
= 0. | ||
| x = 2. | x = −3. | ||||||
See the previous problem.
Next Lesson: Removing grouping symbols
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