If at some point we get a $0$ remainder then the decimal representation is finite otherwise the remainder has to repeat and lead to a repeating non-terminating decimal representation. However, I did some additional reading and came across Cantor's transfinite numbers. Integers are numbers that don't have to be represented as a fraction or a decimal and a subset of integers are whole numbers, which are non-negative integers. This happens because the decimal representation is obtained via division of $p$ by $q$ and hence the only possible choices for remainder are $0, 1, 2,\ldots, q - 1$. Rational numbers are represented as a fraction of two integers, while irrational numbers cannot be represented as a fraction of two integers.
Here the following result helps:Ī rational number $p/q$ can be represented as a finite decimal in base $b$ notation, if and only if denominator $q$ divides $b^$ for some positive integer $n$.Īlso it should be noted that in case the decimal representation is not finite, then it has to follow a repeating pattern. Also it is better to understand why some rationals can have finite decimal representation and others don't have such finite decimal representation. All the integers are included in the rational numbers, since any integer z z can be written as the ratio z 1 z 1. For example, the fractions 13 1 8 1111 8 are both rational numbers. It has to be understood very clearly that a rational number may or may not have finite representation depending on the kind of representation chosen. The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the integer 5 can be written as 5/. Integers are numbers that don't have to be. All integers are rational numbers is true, because every integer can be expressed as a fraction with denominator equal to 1. Rational numbers are represented as a fraction of two integers, while irrational numbers cannot be represented as a fraction of two integers.
These categories include rational numbers, irrational numbers, integers, and whole numbers. Every integer can be expressed as a fraction with a denominator of 1. There are four categories in which numbers can be claified in. Also, 3 is a rational number since it can be written as 3 3 1 and 4.5 is a rational number since it can be written as 4.5 9 2. This means that 2 5 is a rational number since 2 and 5 are integers. Similarly "one/three" can be written as a finite decimal in ternary, but as an infinite one in normal base ten. Integers are a subset of rational numbers. A rational number is any number that we can write as a fraction a b of two integers (whole numbers or their negatives), a and b. A fraction like "one/two" can be written as $0.5$ in decimals (as a finite expression), but the same can't be written as a finite decimal in ternary. In decimal notation the number "five" is written as $5$, but in binary it is written as $101$ and in ternary as $12$. Examples of rational numbers include the following. Now you can see that numbers can belong to more than one classification group. Every integer is a rational number and, but not all. Rational Numbers Rational numbers have integers AND fractions AND decimals. Key Points The set of rational numbers, written, is the set of all quotients of integers.
¯ 3 is rational because this number can be written as the ratio of 16 over 3, or 16 3. Integers are -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on. However the concept of a number is different from the concept of representing it. Rational numbers are numbers that can be written as a ratio of two integers. This confusion is primarily due to the fact that most people try to think of a number and its representation as one and the same thing. For example, $0.33\overline.I believe the fundamental problem (or confusion) here is that OP finds it difficult to believe that a rational number, which is a ratio of two finite integers, can have a representation which is infinite. Standard 8.NS.1 requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of choice of representation.
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