 S k i l l
i n
A L G E B R A

5

# RECIPROCALS

AND

## ZERO

The definition of reciprocals

TWO NUMBERS ARE CALLED reciprocals of one another if their product is 1.

 The reciprocal of 2, for example, is 12 -- because
 2· 12 =  1.
 In general, the symbol for the reciprocal of any number  a  is 1a .
 a· 1a = 1a ·  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· 1a = 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!

 1z
 Problem 2.   If pq = 1, then q = 1p .   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.

 Prove that the reciprocal of 34 is 43 .
 34 · 43 = 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 ab is ba .

Problem 5.

 a) = 2

1 over any number means: The reciprocal of that number.

 b) = 32

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.

 ab =  a· 1b = 1b · 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:  ab = a + (−b).  (Lesson 3.)

In a sense, we do not need the word "division." We could give the purely formal definition:

 This symbol ab means a · 1b

Problem 7.   Rewrite each of the following according to The Definition of Division.

 a) 34 = 3· 14 b) x2 = x· 12 or ½x c) x + 1x + 2 = (x + 1)· 1    x + 2
 d) a + b + c     6 = (a + b + c)· 16
 e) x  ¼ab = 4xab

To divide by ¼ is to multiply by its reciprocal, 4.
This shows how not to leave a fraction in the denominator.

 f) = 3a2b
 Problem 8.   Prove that aa = 1 , for any number a 0.
 aa = a· 1a , according to the Definition of Division, = 1, according to the definition of reciprocals.

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.)

 a) x − 2x − 2 = 1 b) a + b + ca + 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 we start with some number x and then divide it by a number a, then if we wish to preserve x, we have to multiply by a:

 xa · a = x.

Or, if we first multiply by a, then to preserve x we have to divide by a,

 x· a· 1a = x
 or, equivalently, multiply by 1a .
 1a 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 · 13
 b)  27 = 27 3 · 3
 c)   p = pq · q
 d)   m = mn · 1n
 e)   cos x = cos xsin 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 ab

The rule called the Definition of Division --

 ab =  a· 1b
 -- is merely a formal rule.  It tells us that we may replace 12 3 with 12· 13 .

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.

If the quotient of a divided by b is some number n --

 ab = 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.

 0a =  ? a0 =  ? 00 =  ?
 a) 82 = n.   What number is n?

n = 4.  Because according to the meaning of the quotient n, 4· 2 = 8.

 b) 08 = n.  What number is n?

n = 0.  Because 0· 8 = 0.

 Therefore, 0a = 0,  for any number a 0.
 c) 80 = n.  What number is n?

There is no number n such that n· 0 = 8.  Division by 0

 is an excluded operation.  The symbol 80 -- although it
 may look like a number -- is not a number. 80 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) 00 = n.  What number is n?

n could be any number, because for any number n,  n· 0 = 0.

 The symbol 00 could be defined to have any value. We say that

it is indeterminate.

In summary (a 0):

 0a =  0. a0 =  No number. 00 =  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 80 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 00

to be 1, it is possible to define that quotient when x = 2.

 For we can define 00 to be any number.
 However, we can never define the symbol 80 .

Problem 15.

 a) 05 = 0 b) 0  −5 = 0 c) 0x (x 0) = 0
 d) 50 = No number. e) 00 = Any number. f) x0 (x 0) = No number.

Problem 16.   Let x = 2, and evaluate the following.

 a) x − 2x + 2 = 04 = 0. b) x + 2x − 2 = 40 = No value.
 c) 2x − 43x − 6 = 00 = 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 10 .

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 − 2x + 5 =  0. b) x + 3x − 1 =  0. x = 2. x = −3.

See the previous problem. Next Lesson:  Some rules of algebra

Please make a donation to keep TheMathPage online.
Even \$1 will help.