If you are wondering or trying to educate yourself about the differences between rational numbers and irrational numbers, you’ve found the right place! To help you understand math’s rules and terms of rational numbers, we prepared for you some simple examples of rational numbers that can be written as fractions with integers.
Why is it important to recognize rational numbers?
First and foremost, you should know why it is so important to study the art of differentiating rational numbers from irrational numbers. The reason is that it can help you once you begin figuring out more complex mathematics such as algebraic equations. As there are rules that you’ll need to follow in order to get the correct result, you will have to rely on whether or not you’re dealing with a whole rational number. Therefore, it’s great to cement the basics of math such as understanding key mathematics rules and terms to save yourself a headache in the future.
How to define the integer term
To begin with, you will have to know the definition of the integer term to discover a rational number that isn’t an integer. In the study of mathematics, an integer is not a number but a whole number, nor is it a fractional number. For example, numbers 0-10 are integers. In addition, they are also rational numbers as they can be fractions of 0/1, 1/1, 2/1, 3/1, 4/1, 5/1, and so on. Nevertheless, please take note that integers don’t have to specifically be positive whole numbers. They can also be negative. Here are examples of rational integers that are negative numbers: -1, -2. -3. -4. -5 and so on.
How to define a rational number that is not an integer
Once you fully comprehend what an integer is, it will be much easier to find a rational number that isn’t an integer. A rational number is any number that can be written as a fraction with integers. Here are 3 types of rational numbers aside from the integers we previously mentioned above:
A ratio of integers: 1/7, 3/4
Please take note that there is more than one way to write the same rational number as a ratio of integers. For instance, 1/7 and 2/14 are the same ratios of integers.
All mixed numbers: 3 1/5 which can be rewritten as 16/5 (becoming to a ratio of integers)
Decimal numbers (reoccurring and repeating decimal numbers): 0.5, 0.79, 0.36363, 0.444444, 0.242424 and 0.5555
How irrational numbers are created
Even though most of the numbers can be rewritten as a fraction with integers, there are still some numbers that can’t be rewritten as a fraction with integers. That means they are not rational numbers. In fact, mathematicians created some symbols for the numbers that are reoccurring and repeating decimal numbers and called the combination of them the “irrational numbers”. Some most-used examples of the symbols are π (pi) and √(square root). For example, the square root of any prime number is created for the decimal numbers like 1.41421… =√2, 1.7325… = √3, and π are created for the decimal number of 3.14159…. Nevertheless, if Pi is written as the fraction of 22/7 or 3.14, they both are rational numbers. Moreover, do not forget to keep in mind that √4 is a rational number, even though it contains a square root symbol since √4 = 2 (positive rational integer), as well as √9 = 3.
Why is a repeating decimal number considered a rational number?
Now some of you may wonder why repeating decimal numbers are considered rational numbers. The answer is that the repeating decimal numbers can be described as a ratio of two integers. Therefore, any decimal which uses repetitive numbers such as 0.575757 and 0.818181 will always be considered rational numbers as they can be the ratio of 575757/1000000 and 818181/10000000. At the same time, all numbers which repeat themselves continuously such as 0.5555 (5555/100000), 0.33333 (33333/1000000), and 0.8888 (8888/100000) are always rational numbers, which aren’t also in the classification of integers or whole numbers.
For those who may wonder if there is an official mathematical term for infinite decimal numbers, the answer is yes, of course. There is actually an official mathematical term for decimal numbers that are infinite and never end. Any set of numbers that are repeated indefinitely in the decimal form of a rational number can be called by the term repetend. For example, the rational number 17 in its decimal form looks like this: 0.142857142857 where the digits 142857 are the repetend. Lastly, it is important to note that only rational numbers have repetends and that sometimes you will notice a line over the repeating digits. This line is called a vinculum and is only there to indicate the repetend.
We hope your comprehension of integers, rational numbers, and irrational numbers will be useful for your math solving in the near future. Good luck and get an A for us!