Math has become easier than ever now that we have calculators at our fingertips and as part of our phones. But dealing with numbers that mean something is easy. The number 0, due to its multiple properties, can certainly stump a lot of people.
Zero is a rational number because it is an integer. An integer is a number that is complete without containing a decimal or a fraction. Rational numbers are also expressed in numerators and denominators, where both are equal integers. Since 0 does not have a repeating decimal, it is a rational number.
Many people are still scared of words like integers, irrational, and rational numbers. That fear is unfounded. Keep reading as we break down the easiest ways to understand rational and irrational numbers.
What are Rational and Irrational Numbers?
A rational number is a number that we can write as a ratio of two integers, otherwise known as a fraction (source). Rational numbers can be both positive or negative.
Numbers that have a non-repeating decimal can also be a rational number. Whole numbers, integers, and perfect square roots are all examples of rational numbers.
Irrational numbers are numbers that are not integers. Any number that contains a decimal point that is continuous and non-repeating, such as (pi), which is 3.14159…, is an irrational number.
There are several tests and tools that you can use to determine if a number is rational or irrational.
Identifying Rational and Irrational Numbers
The main point about rational numbers is that within the fraction, the numerator (top number) and the denominator (bottom number) should both be integers. The focus of rational numbers is not on rationality, but on the word “ratio.”
Types of Rational Numbers
You will encounter several types of numbers that can be rational, including whole numbers and integers, decimals, and fractions.
Whole Numbers and Integers
The easiest types of rational numbers are whole numbers. Whole numbers are sometimes called integers, but they are slightly different. Whole numbers refer to positive numbers that do not contain a fraction or decimal point. Examples of whole numbers are 1, 15, 3900, 6245, etc.
Integers, on the other hand, incorporate whole numbers and also their negative counterparts. So the number 3 is both a whole number and an integer. The number -3 is not a whole number, but it is an integer (source).
Fractions are rational numbers because the very nature of rational numbers involves ratios. Since fractions have both a numerator and a denominator, they are rational numbers.
Since a fraction can also be written as a decimal, decimals are also an example of a rational number. The vital distinction here is that the decimal can either repeat or terminate. For example, the fraction 1/3rd can also be represented as the decimal .3, repeating.
Another example is the fraction 9/10, which can also be represented as 0.9 and is a terminating decimal — it does not repeat.
A repeating decimal refers to the numbers after the decimal that would repeat with the same number or with the same set of numbers to infinity (source).
This does not always have to be the first number after decimal that repeats. For example, in the fraction of 1/11, you would get the repeating decimal of 0.090909.
A terminating decimal refers to the decimal that does not repeat. A sign of this is when the number ends with a 0. For example, the fraction of 1/2 is represented as the decimal .50 or .5.
Because it is exactly half, it does not need to repeat, and it is referred to as a terminating decimal.
There are also several methods that you can use to determine if a number is rational or irrational — we’ll get into these strategies next.
Ask yourself, “Is the number a whole number, an integer, a fraction, or does it contain a repeating or terminating decimal?”
If your answer is yes, you have a rational number on your hands. While this method is the easiest, it does require knowledge of some of these concepts. If you are unsure or you are out of practice, then using a testing method might work better for you.
One trick that some mathematicians use to identify rational numbers is the power of your imagination. You would not generally associate imagination with math, but a mathematician might say that if a number is easy to imagine, it is quite likely a rational number.
While we might consider word problems to be grade school concepts, the visual conceptualization of elements is vital and very helpful in mathematics.
If you imagine Jane with 20 apples and John with 15, quantifying these numbers as physical objects will make it simpler to understand.
In the case of fractions, imagining 1/2 a pie, or a quarter or 1/5th allows you to calculate easy fractions in your head, and it works up a healthy appetite for dessert, too.
Therefore, any number or concept that you can easily imagine is generally a rational number because no one is thinking of 23.4658742 percentage of a pie, but you certainly can imagine 25% of a pie.
It is also easy to imagine an empty pie plate, and since you can imagine the number 0 as the absence of a number, 0 is a confirmed rational number using this method.
A quick way to determine if a number is rational is by doing the ratio test. We’ll take a look at an example using the number 1.
1 = 1/1 = 23/23 = 4500/4500
One is a versatile number and can be represented in an infinite number of ratios, as all of them equal one. However, this test can be repeated with any integer. If a number contains a decimal, it cannot pass the ratio test, and hence, it would not be a rational number.
Even incredibly large numbers can be rational numbers if they pass the ratio test. As long as the number is used as a numerator over the denominator “1” and gives back the exact same result, it is a rational number.
Here are a few more examples:
6/1 34/1 100/1 78600/1
Since all of these numbers are divisible by 1 and get the same result as the original numerator, they are all rational numbers.
When looking at numerators and denominators of a rational number, numerators can be any number, while denominators can be any integer except 0. Since 0/1 is still 0, this confirms that 0 is a rational number using this strategy (source).
Another method is to check whether a square root is a perfect number or an imperfect number. Most results of square roots are irrational numbers, but the result of a perfect square root is a whole number, and hence, these are also rational numbers.
A square root is a number that has a specific result when multiplied by itself. For example, 2 is the square root of 4. That means when 2 is multiplied by itself, you will get 4, or 2 = √4.
A few other examples of perfect square roots are √1 is equal to 1, √9 is equal to 3, √16 is equal to 4, and √25 is equal to 5.
As these numbers create perfect whole numbers, perfect square roots are rational numbers.
In comparison, other numbers will not have perfect square roots. For example, the √15 is 3.87298334621 and contains a non-repeating and continuous decimal, making it an irrational number.
Since the √0 is equal to 0, 0 is again confirmed as a rational number using all 3 methods.
Positive and Negative Rational Numbers
When discussing rational numbers, there are both positive and negative rational numbers. Since integers indicate rational numbers, any point on a number line is an acceptable rational number (source).
When discussing positive and negative numbers, it is best to imagine these numbers on a number line. The number 0 would be in the middle of the line, and all positive numbers appear on the right, while all negative numbers appear on the left.
Positive numbers go up numerically from 1 to 2 to 3 and so forth. Of course, there is also room for decimals in between whole numbers. Negative numbers count down similarly, from -1 to -2 to -3 and so forth.
Positive rational numbers are simply expressed in their normal form as an integer, fraction, or decimal. Negative rational numbers must include a minus sign before the number to indicate its placement on a number line compared to a positive number.
Properties of 0
This brings us to the number 0. Since 0 is the absence of anything, it is probably the easiest concept to understand.
There is some debate about whether 0 is a rational number. Since 0 cannot be divided by 0 or represented as 0/0, some people argue that this makes it an irrational number, as it does not meet all the requirements for a rational number (source).
However, Siri will sarcastically remind you if you ask what 0 divided by 0 is — imagine that you are trying to share zero cookies with zero friends, there are no cookies and no friends. Zero cannot be divided by zero, which makes it undefinable rather than irrational.
On the topic of 0/0, you can expect 1 of 3 results. First, if a number is divided by itself, it should equal one, as shown in the calculations previously. Second, 0 divided by anything should still be 0. Third, if you input 0 divided by 0 on a calculator, it will remind you that it is not calculable.
Since 0 is not a negative number, it is both a whole number and an integer. It might not meet every requirement of a rational number, but it does not meet any requirements of an irrational number.
Interestingly enough, zero is the number that is most intuitively understood by humans. It is the “no” of the mathematical world. In the first few years of our academic life, we only dealt with 0 and positive numbers.
Many students struggle with the concept of negative numbers since we cannot imagine less than 0. Zero also has some other interesting qualities. For example, 0 is an even number.
Zero is also the only number which is not positive or negative, and it is also the only number which is a multiple of all numbers, even though the result will always be 0.
Zero is also known as the identity element, which means that as soon as any number is added to zero, the original number is the sum total. Zero is also not a prime number or a composite number, as it is not required in the production of itself.
Zero is also an adjective, like all other numbers when they are used to modify a noun. But there is a lot more to numbers being used as adjectives, so you’ll want to read “Are Numbers Adjectives?” to ensure you always use numbers correctly when writing.
Finally, if two numbers are added together and the result is 0, the two numbers added together are called additive inverses and opposites. An example of this is -5 + 5 = 0. Only zero can give numbers this quality.
Positive numbers and 0 intrinsically fit into our worldview of something which is tangible, in the case of positive numbers, and something which is intangible and absent, in the case of 0.
While positive numbers have to be added to something physical to make them tangible, 0 is a number that exists on its own.
This very nothingness makes 0 a unique and powerful number in multiple fields, from mathematics to coding to electronics. Ultimately, it may be the most rational number of all.