|lesson 5.1 rational expressions|
In common terms, rational numbers are fractions.
Look at the first five letters in the word rational. They spell ratio and that's exactly what rational numbers are.
definition The set of rational numbers denoted Q (for quotient) is defined to be
the set of all numbers represented by
where a, the numerator and b, the denominator Z (the set of integers) and b 0.
Now, let's discuss the issue of b = 0. Our number system is constructed such that division by zero is undefined and/or equal to infinity. This is because of the fact that the smaller the denominator or divisor, the larger the quotient.
When we divide 8 by 4, we get 2. When we divide 8 by 2, we get 4. If we divide 8 by 0.000002, we get 4,000,000, so imagine how large the answer will be when we divide 8 by 0.
Really, zero is just a place holder half way between 1 and 1. This is why we define a rational number to be a fraction whose denominator is not equal to zero. Thus, whenever we are working with rational numbers in algebra, we must specify that the denominator is not equal to zero.
For instance, say we are asked to solve . The very first thing we will specify in our solution is that x 5 since x = 5 would make our denominator equal to zero.
Any integer (whole number) such as 5, is a fraction because we can write it as . This often causes confusion for students who are doing a question such as: , which of course really means you should multiply the numerator by 2 and the denominator by 1 like this: .
The denominator of an integer is 1.
With rational numbers or expressions just follow the rules and you'll get the right answers.
|multiplication & division||change of sign||examples||practice||solutions|
multiplication and division of rational numbers
Fraction facts about multiplication and division
Let's discuss each fact in the list:
a) means that a single negative can be placed anywhere in the fraction.
b) means that the negative exponent, flips the fraction.
c) means that integers have a denominator of 1.
d) tells us how to multiply fractions -- multiply the tops and multiply the bottoms.
e) means that 0 multiplying anything except an indeterminate form will = 0.
f) means that division by 0 yields a huge value, which we call infinity or undefined.
g) means that division by a fraction is the same as multiplication by its reciprocal.
This is not only true for fractions. This is true for all numbers.
If you want to divide something by 2, you take half of it!
Since division is the inverse of multiplication, dividing by a is the same as multiplying by , no matter what kind of number "a" is.
Equivalent Fractions or Reducing
Reducing (once referred to as canceling) is one of the least understood operations in math.
The most common error is reducing across addition or subtraction.
For example, many students will incorrectly perform the following operation:
because they incorrectly reduce or cancel the b's.
If we do this example with integers, it becomes obvious that the result is not at all accurate
since we know that
So what is reducing?
Reducing, a technique used to simplify fractions in order to express them in lowest terms, is simply the act of rewriting a fraction as an equivalent value because there is a factor of 1 in the original expression of the fraction.
As you can see from the last example, the simplification of fractions or rational numbers is easily accomplished when both numerator and denominator are in factored form.
Be kind to yourself as you work with rational numbers. If your first step is to factor wherever possible, you'll have a lot less grief from fractions. Remember that reducing requires there be the same factor in a numerator and denominator.
|intro||change of sign||examples||practice||solutions|
|Reduce like terms in numerator and denominator.|
|Factor, Reduce like terms,|
|Invert the 2nd fraction.Factor.|
|, x y||Reduce like terms and multiply.|
|Invert the 3rd fraction.|
|Factor. Reduce and multiply.|
|intro||multiplication & division||change of sign||practice||solutions|
change of sign
To illustrate this process, let's look at an example.
First, we factor both the numerator and the denominator to get
The problem now is that (x 5) is not equal to (5 x), so we can't reduce.
However, if we factor 1 from the (5 x), we will get (x 5), and then we can reduce.
The problem therefore can be solved by this process as shown.
Though we call this change of sign, we're not really changing any signs or values. We're simply expressing the terms in a different form to suit our purpose.
We're using the fact that a x = (x a).
Perform the following operations
|Factor -1 out of denominators.|
Factor the trinomial.
|Reduce and multiply|
|b)||Invert the divisor.Factor the denominator.|
Factor -1 in numerator and denominator of the 1st fraction.
Factor the numerator of the 2nd fraction.
|intro||multiplication & division||examples||practice||solutions|
Perform the indicated operations. Reduce to lowest terms.
|intro||multiplication & division||change of sign||examples||solutions|
|intro||multiplication & division||change of sign||examples||practice|
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