Most of us undergo years of faculty 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 will not be a explanation for frustration in any respect, in the event that they had been taught fairly and clearly.
Sadly most of these items are speculated to be lined in elementary faculty, and most simple faculty academics should not have an excellent understanding of fundamental math ideas. As an alternative they’re supposedly to show only a assortment of "expertise."
One of many easiest ideas that’s often left inadequately defined is the distinction between fractions and rational numbers. Let's see if we are able to clear it up now.
A fraction is a quantity that expresses half of an entire as a quotient of integers (the place the denominator just isn’t zero).
A rational quantity is a quantity that may be expressed as a quotient of integers (the place the denominator just isn’t zero), or as a repeating or terminating decimal. Each fraction suits the primary a part of that definition. Subsequently, each fraction is a rational quantity.
However regardless that each fraction is a rational quantity, not each rational quantity is a fraction.
Why? Think about 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 corresponding to four or 1 could be expressed because the quotient of integers.
However an integer just isn’t a fraction . four is an integer, however it isn’t a fraction. four just isn’t expressed because the quotient of integers. The distinction right here is within the wording.
A fraction is a quantity that expresses half of an entire. An integer doesn’t categorical a component. It solely expresses a complete quantity.
A rational quantity is a quantity that could be expressed as a quotient of integers, or as half of an entire, however fraction is a quantity that is (have to be) expressed as a quotient of integers, or as half of an entire – 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 just isn’t a fraction, as a result of an integer just isn’t a ratio. It could be expressed as a ratio, but it surely just isn’t a ratio in itself; it can be divided by one other complete quantity, but it surely i s not being divided.
In a nutshell, the fractions are a subset of the rational numbers. The rational numbers comprise the integers, and fragments don’t.