Most of us undergo years of college math programs and nonetheless are confused about some basic items. For instance: Why cannot you divide by zero? Why is .999… equal to 1, and never a bit much less?
There are a great deal of these sorts of questions, that would not be a reason for frustration in any respect, in the event that they have been taught fairly and clearly.
Sadly most of these items are purported to be coated in elementary college, and most basic college academics do not have a very good understanding of primary math ideas. As a substitute they’re supposed to show only a assortment of “abilities.”
One of many easiest ideas that’s often left inadequately defined is the distinction between fractions and rational numbers. Let’s examine if we will clear it up now.
A fraction is a quantity that expresses half of a complete as a quotient of integers (the place the denominator shouldn’t be zero).
A rational quantity is a quantity that may be expressed as a quotient of integers (the place the denominator shouldn’t be zero), or as a repeating or terminating decimal. Each fraction matches the primary a part of that definition. Subsequently, each fraction is a rational quantity.
However although each fraction is a rational quantity, not each rational quantity is a fraction.
Why? Contemplate this:
Each integer (all the entire numbers, together with zero, and their negatives….-Three, -2, -1, zero, 1, 2, Three…) is a rational quantity, as a result of it may be expressed as a quotient of integers, as within the case of four = eight/2 or 1 = Three/Three or -Three = Three/-1 and so forth. So integers akin to four or 1 will be expressed because the quotient of integers.
However an integer shouldn’t be a fraction. four is an integer, however it isn’t a fraction. four shouldn’t be expressed because the quotient of integers. The distinction right here is within the wording.
A fraction is a quantity that expresses half of a complete. An integer doesn’t specific a component. It solely expresses an entire quantity.
A rational quantity is a quantity that will be expressed as a quotient of integers, or as half of a complete, however fraction is a quantity that is (have to be) expressed as a quotient of integers, or as half of a complete – there’s a distinction. The distinction is delicate, however it’s actual.
There are barely totally different variations of the definition of a fraction, together with, “A fraction is the ratio of two complete numbers, or to place it merely, one complete quantity divided by one other complete quantity.”
That definition additionally exhibits that an integer shouldn’t be a fraction, as a result of an integer shouldn’t be a ratio. It will be expressed as a ratio, but it surely shouldn’t be a ratio in itself; it can be divided by one other complete quantity, but it surely is not being divided.
In a nutshell, the fractions are a subset of the rational numbers. The rational numbers comprise the integers, and fractions do not.