Greetings everyone! I have a quick question that involves numbers and operations. Can all repeating numbers be classified as a rational number or not?
If by repeating numbers you mean numbers that have a repeating sequence of decimals, the answer is yes, they can all be represented by a fraction, thus they are rationals.
e.g. 0.67321321321.... = 11209/16650
e.g. 3.181818... = 3 + 18/99 = 3 + 2/11 = 35/11
I am truly iconic!
Hello! That's a great question. Yes, all repeating numbers can be classified as rational numbers. Before we dive into the explanation, let me clarify what rational numbers are. Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both whole numbers.
Now, let's consider repeating numbers. A repeating number is a decimal that has a repeating pattern, denoted by a bar over the repeating part. For example, the decimal representation of the fraction 1/3 is 0.3333..., with a bar over the threes to indicate the repeating pattern.
To prove that repeating numbers are rational, we can use a mathematical technique called algebra. Let's take an example of a fraction, say 1/7, and represent it as a decimal. The decimal representation is 0.142857142857..., with a bar over the repeating pattern "142857".
To convert this repeating decimal into a fraction, we can use a clever trick. Let's call the repeating part (in this example, "142857") as "x". By multiplying the decimal representation by a power of 10, we can align the repeating part with the original decimal. In this case, multiplying by 1000000 gives us 1000000x = 142857.
Now, let's solve for "x". Dividing both sides by 1000000 gives us x = 142857 / 1000000. Simplifying the fraction, we get x = 1/7. Therefore, the repeating decimal 0.142857... is equal to 1/7, which is a fraction and hence a rational number.
By using the same technique, any repeating decimal can be converted into a fraction, proving that all repeating numbers are rational numbers.