They can be written as a ratio of two integers. This article concentrates on rational numbers. Clearly all fractions are of that form, so fractions are rational numbers. Terminating decimal numbers can also easily be written in that form:
So, for example, 0. But do you know why this is true? Despite what your teachers may have told you, the most important question in mathematics is not how, it is why!
Today I will show why every rational number has a terminating or eventually repeating decimal expansion, and in a future post I will show why every repeating or terminating decimal expansion represents a rational number. First, what is a decimal expansion?
By convention, we use place notation to give each digit a value based on a negative power of ten. For example, recalling thatNow, how can we convert a rational number, represented as a fractioninto a decimal representation?
The answer, of course, is to use long division to divide p by q, extending p with zeros after the decimal point as necessary. Consider performing the division. At each step in the division, we are left with a remainder. This remainder, of course, will always be a value between 0 and q-1inclusive if the remainder were larger, it would mean that the quotient we wrote on top was too small.
If the remainder is ever zero, then we are done: If we never get zero as a remainder, the fact that there are only a finite number of possible remainders means that at some point, we must get a remainder that we already got before.
But this means that the quotient will now repeat: It should be clear from the example that after reaching a remainder of 3 for the second time that the quotient will start repeating: As a bonus challenge, can you figure out how to tell the difference between rational numbers whose decimal expansion terminates, and those whose expansion repeats?
Note that we also know something else: You can also look at this in reverse; for example, if you see a periodic decimal whose repeating portion is ten digits long, you know that the rational number it represents must have a denominator of at least Maybe you can try to figure this out for yourself in the meantime!Terminating Decimal While expressing a fraction in the decimal form when we perform division we notice that the division is complete after a certain number of steps i.e., we get the remainder zero.
The quotient obtained as decimal is called the terminating decimal. A repeating decimal is not considered to be a rational number it is a rational number. We have different ways of representing numbers, for example the number of fingers on my left hand can be represented by the English word five, or the French word cinq or the symbol 5 or the Roman numeral V or the fraction 10/2 or many other ways.
To convert a Decimal to a Fraction follow these steps: Step 1: Write down the decimal divided by 1, like this: decimal 1 Step 2: Multiply both top and bottom by 10 for every number after the decimal point.
(For example, if there are two numbers after the decimal point, then use , if there are three then use , etc.). Many people are surprised to know that a repeating decimal is a rational number.
The venn diagram below shows examples of all the different types of rational, irrational nubmers including integers, whole numbers, repeating decimals and more.
Jan 26, · The proof of this lies in the fact that every terminating decimal has the form n/10^e, for some e >= 0 (e is the number of places to the right that the decimal point must be moved to give you an integer, and n is that integer), and every fraction of that form has a terminating decimal found by writing down n and moving the decimal point e.
Write the decimal number as a decimal fraction and reduce the fraction to lowest terms. is a non-terminating repeating decimal. For example, when you divide 16 by 3, the result Decimals, and Fractions Technique.