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Rational numbers
consider the two rational numbers as
and
.
What would we get if we add these two rational numbers, i.e. what is the value of
This means that the sum of two rational numbers
and
?
is a rational number. In other words, we
can say that rational numbers are closed under addition.
Is this true for all rational numbers?
Yes. We can try for different rational numbers and see that this property is true for all rational
numbers. Thus, we can say that the sum of two rational numbers is again a rational number. In other
words, we can say that rational numbers are closed under addition. This property of rational
numbers is known as the closure property for rational numbers and it can be stated as follows.
“If a and b are any two rational numbers and a + b = c, then c will always be a rational
number”.
Are rational numbers closed under subtraction also?
Let us find out.
Consider two rational numbers
and
.
Thus, rational numbers are closed under subtraction also.
Closure property of rational numbers under subtraction can be stated as follows.
“If a and b are any two rational numbers and a − b = c, then c will always be a rational
number”.
Now, let us check whether rational numbers are closed under multiplication also. For this, consider
two rational numbers
Now,
and
.
, which is a rational number.
Thus, rational numbers are closed under multiplication also.
Closure property of rational numbers under multiplication can be defined as follows.
“If a and b are any two rational numbers, then a × b = c, then c will always be a rational
number”.
But rational numbers are not closed under division. If we consider the division of
, then we will
not obtain a rational number.
is not a rational number because division of a rational number by zero is not defined.
Thus, we can say that rational numbers are not closed under division.
We can summarize the above discussed facts as follows.
•
Rational numbers are closed under addition, subtraction and multiplication.
•
Rational numbers are not closed under division.
Consider the rational numbers
.
What will be the values of the expressions
?
Let us see.
Thus,
Here we see that the values of both the expressions are same. This is the distributive property of
rational numbers for multiplication over addition. This property is true for all rational numbers and
mathematically it can be written as follows.
If x, y and z are any three rational numbers, then x × (y + z) = (x × y) + (x × z).
Does this distributive property hold for multiplication over subtraction also?
Let us check it.
Consider the rational numbers
.
Let us find the value of the expressions
Thus,
Hence, we notice that the result is same from both the ways. Thus, the distributive property of
rational numbers for multiplication over subtraction can be written as follows.
“If x, y, and z are any three rational numbers, then x × (y – z) = (x × y) – (x × z)”.
Let us now look at some more examples.
Example 1:
Find the value of the following expression using appropriate properties.
Solution:
[By commutative property for addition]
[By commutative property for multiplication]
[By distributive property of multiplication over addition]
Example 2:
Solve the following expression.
Solution:
[By commutative property for addition]
[By distributive property of multiplication over subtraction]
Exponents
Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the
shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals"
3
sign in (5)(5)(5) = 5 . The "exponent", being 3 in this example, stands for however many times the value
is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".
This process of using exponents is called "raising to a power", where the exponent is the "power".
"p
The
3
expression "5 " is pronounced as "five, raised to the third power" or "five to the third". There are two
specially-named
named powers: "to the second power" is generally pronounced as "squared", and "to the third
3
power" is generally pronounced as "cub
"cubed". So "5 " is commonly pronounced as "five cubed".
3
When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "3 ". But with
6
variables, we need the exponents, because we'd rather deal with "x " than with "xxxxxx".
Exponents have a few rules that we can use for simplifying expressions.
•
3
4
Simplify (x )(x )
Copyright © Elizabeth S2011 All Rights Reserved
To simplify this, I can think in terms of what those exponents mean. "To the third" means
"multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can
"expand" the two factors, and then work backwards to the simplified form:
(x3)(x4) = (xxx)(xxxx
xxxx)
= xxxxxxx
= x7
7
(3+4)
Note that x also equals x
. This demonstrates th
the
e first basic exponent rule: Whenever you multiply
two terms with the same base, you can add the exponents:
(x
m
n
)(x )=x
(m+n)
4
However, we can NOT simplify (x
Nothing combines.
•
)(
)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3).
2 4
Simplify (x )
Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the
2
fourth" means that I'm multiplying four copies of x :
(x2)4 = (x2)(x2)(x2)(
)(x2)
= (xx)(xx)(xx
xx)(xx)
= xxxxxxxx
= x8
8
( 2×4 )
Note that x also equals x
. This demonstrates the second exponent rule: Whenever you have an
exponent expression that is raised to a power, you can multiply the exponent and power:
m
n
(x ) =x
mn
If you have a product inside parentheses, and a power on the parentheses, then the power goes on each
2 3
2
2
2
2 2 2
3 6
element inside. For instance, (xy ) = (xy )(xy )(xy ) = (xxx)(y y y ) = (xxx)(yyyyyy)) = x y =
(x)3(y2)3. Another example would be:
Warning: This rule does NOT work if you have a sum or difference within the parentheses. Exponents,
unlike mulitiplication, do NOT "distribute
distribute" over addition.
2
2
2
For instance, given (3 + 4) , do NOT succumb to the temptation to say "This equals 3 + 4 = 9 + 16 =
25", because this is wrong. Actually, (3 + 4)2 = (7)2 = 49, not 25. When in doubt, write out the
2
expression according to the definition of the power. Given (x – 2) , don't try
ry to do this in your head.
2
2
Instead, write it out: "squared" means "times itself", so (x – 2) = (x – 2)(x – 2) = xx – 2x – 2x + 4 = x
– 4x + 4.
The mistake of erroneously trying to "distribute" the exponent is most often made when the student is
trying to
o do everything in his head, instead of showing his work. Do things neatly, and you won't be as
likely to make this mistake.
There is one other rule that may or may not be covered at this stage:
Anything to the power zero is just "1".
This rule is explained
d on the next page. In practice, though, this rule means that some exercises may be
a lot easier than they may at first appear:
•
4 7 12 5
9 3 4 2 0
Simplify [(3x y z ) (–5x y z ) ]
Who cares about that stuff inside the square brackets? I don't, because the zero power on the
th outside
means that the value of the entire thing is just 1. A negative exponent just means that the base is on the
–2
wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x " (ecks to
2
2
the minus two) just means "x , but underneath, as in 1/(x )"
–4
•
Write x using only positive exponents.
•
Write x / x using only positive exponents.
•
Write 2x using only positive exponents.
2
–3
–1
Note that the "2" above does not move with the variable; the e
exponent is only on the "x".
•
–2
Write (3x) using only positive exponents.
Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the
three as well as the variable.
•
–2
–3 –2
Write (x / y ) using only positive exponents.
This one can also be done as:
Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
Since exponents indicate multiplication, and since order doesn't matter in multiplication, there will often be
more than one sequence of steps that will lead to a valid simplification of a given exercise. Don't worry if
the steps in your homework look quite different from the steps in a classmate's homework. As long as
your steps were correct, you should both end up with the same answer
answer.
By the way, now that you know about negative exponents, you can understand the logic behind the
"anything to the power zero" rule:
Anything to the power zero is just "1".
Why is this so? There are various explanations. One might be stated as "because tthat's
hat's how the rules
work out." Another would be to trace through a progression like the following:
35 = 36 ÷ 3 = 36 ÷ 31 = 36––1 = 35= 243
34 = 35 ÷ 3 = 35 ÷ 31 = 35––1 = 34= 81
33 = 34 ÷ 3 = 34 ÷ 31 = 34––1 = 33= 27
32 = 33 ÷ 3 = 33 ÷ 31 = 33––1 = 32= 9
31 = 32 ÷ 3 = 32 ÷ 31 = 32––1 = 31= 3
Then logically 3
0
= 31 ÷ 31 = 31–1 = 30 = 1.
A negative-exponents
exponents explanation of the "anything to the zero power is just 1" might be as follows:
m0 = m(n – n) = mn × m–n = mn ÷ mn = 1
...since anything divided by itself is just "1".
0
Another comment: Please don't ask me to "define" 0 . There are at least two ways of looking at this
quantity:
•
•
0
Anything to the zero power is "1", so 0 = 1.
0
Zero to any power is zero, so 0 = 0.
0
0
As far as I know, the "math gods" have not yet settled on a "definition" of 0 . In fact, in calculus, "0 " will
be called an "indeterminate form". If this quantity comes up on class, don't assume: ask your instructor
what you should do with it.
By using exponents, we can reformat numbers. For very large or very small numbers, it is
sometimes simpler to use "scientific notation" (so called, because scientists often deal with very
large and very small numbers).
The format for writing a number in scientific notation is fairly simple: (first digit of the number)
followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an
appropriate power). The conversion is fairly simple.
Write 124 in scientific notation.
This is not a very large number, but it will work nicely for an example. To convert this to
scientific notation, I first write "1.24". This is not the same number, but (1.24)(100) = 124 is, and
100 = 102. Then, in scientific notation, 124 is written as 1.24 × 102.
Actually, converting between "regular" notation and scientific notation is even simpler than I just
showed, because all you really need to do is count decimal places.
Write in decimal notation: 3.6 × 1012
Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need
to move the decimal point to the right, in order to make the number LARGER. Since the
exponent on 10 is "12", I'll need to move the decimal point twelve places over. First, I'll move
the decimal point twelve places over. I make little loops when I count off the places, to keep
track:
Then I fill in the loops with zeroes: Copyright © Elizabeth Stapel 2000-2011 All Rights
Reserved
In other words, the number is 3,600,000,000,000, or 3.6 trillion
Idiomatic note: "Trillion" means a thousand billion — that is, a thousand thousand million — in
American parlance; the British-English term for the American "billion" would be "a milliard", so
the American "trillion" (above) would be a British "thousand milliard".
Write 0.000 000 000 043 6 in scientific notation.
In scientific notation, the number part (as opposed to the ten-to-a-power part) will be "4.36". So I
will count how many places the decimal point has to move to get from where it is now to where
it needs to be:
Then the power on 10 has to be –11: "eleven", because that's how many places the decimal point
needs to be moved, and "negative", because I'm dealing with a SMALL number. So, in scientific
notation, the number is written as 4.36 × 10–11
Convert 4.2 × 10–7 to decimal notation.
Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a
seven, I will be moving the decimal point seven places. Since I need to move the point to get a
small number, I'll be moving it to the left. The answer is 0.000 000 42
Convert 0.000 000 005 78 to scientific notation.
This is a small number, so the exponent on 10 will be negative. The first "interesting" digit in
this number is the 5, so that's where the decimal point will need to go. To get from where it is to
right after the 5, the decimal point will need to move nine places to the right. Then the power on
10 will be a negative 9, and the answer is 5.78 × 10–9
Convert 93,000,000 to scientific notation.
This is a large number, so the exponent on 10 will be positive. The first "interesting" digit in this
number is the leading 9, so that's where the decimal point will need to go. To get from where it is
to right after the 9, the decimal point will need to move seven places to the left. Then the power
on 10 will be a positive 7, and the answer is 9.3 × 107
Just remember: However many spaces you moved the decimal, that's the power on 10. If you
have a small number (smaller than 1, in absolute value), then the power is negative; if it's a large
number (bigger than 1, in absolute value), then the exponent is positive.
Warning: A negative on an exponent and a negative on a number mean two very different things!
For instance:
–0.00036 = –3.6 × 10–4
0.00036 = 3.6 × 10–4
36,000 = 3.6 × 104
–36,000 = –3.6 × 104