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 would not be a reason for frustration in any respect, in the event that they have been taught fairly and clearly.
Sadly most of this stuff are purported to be coated in elementary college, and most simple college academics do not have understanding of primary math ideas. As an alternative they’re supposed to show only a assortment of “expertise.”
One of many easiest ideas that’s normally left inadequately defined is the distinction between fractions and rational numbers. Let’s examine 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 matches 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 equivalent to four or 1 may be expressed because the quotient of integers.
However an integer just isn’t a fraction. four is an integer, however it’s not 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 specific an element. It solely expresses a complete quantity.
A rational quantity is a quantity that may be expressed as a quotient of integers, or as half of an entire, however fraction is a quantity that is (should 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 completely different variations of the definition of a fraction, together with, “A fraction is the ratio of two entire numbers, or to place it merely, one entire quantity divided by one other entire 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 may be expressed as a ratio, however it just isn’t a ratio in itself; it can be divided by one other entire quantity, however it 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.