Video: Properties of Rational Number

Is every rational number an integer?

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Video Transcript

Is every rational number an integer?

In this question, we have two very mathematical terms, rational and integer. Letโ€™s take the word integer first. This will be defined as a number that has no a fractional part. It includes the counting numbers, for example, one, two, three, four, zero, and the negatives of the counting numbers. A rational number is defined as one, which can be expressed as ๐‘ over ๐‘ž where ๐‘ and ๐‘ž are integers and ๐‘ž is not equal to zero.

So letโ€™s take some numbers which are rational. For example, we could have this fraction six over one, which fits the form of a rational number. Six and one are integers and the one, this ๐‘ž-value on the denominator, is not equal to zero. This would be equivalent to the value six, which is an integer. Letโ€™s take another rational number. Here, we have the example negative two-fifths. We should ask ourselves if negative two-fifths is an integer, a number that has no fractional part. And this would be no. We couldnโ€™t write this in any way as a number that has no fractional part. Therefore, the answer to the question โ€œis every rational number an integer?โ€ is no.

If we consider the Venn diagram where we have the set of rational numbers, then contained within this set will be the set of integers. Using this diagram is helpful to illustrate that every integer is a rational number, but not every rational number is an integer.

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