Rational numbers
Introduction to Rational Numbers
So far you have studied about fractions, decimals, integers and even percentages. Well, now it’s time to expand your knowledge and understanding of the number system even further. Rational numbers have ended up being a challenge for the brightest of students because of the way they have been presented to them since the dawn of time. At Cuemath, we seek to change the dogmatic approach and utilize what has already been learnt to introduce new concepts.
Rational Numbers are numbers that can be represented using integers. More precisely, if you apply the four basic arithmetic operations (addition, subtraction, multiplication, division) to integers in any order, you will get rational numbers. For example:
\( \frac{{\left( {3 + 7} \right)}}{{11}} = \frac{{10}}{{11}},\frac{{\left( {2 - 5 \times 3} \right)}}{{26}} = - \frac{1}{2}\)
✍Note: Any integer is itself a rational number.
What are Rational Numbers?
Any number that can be represented in the form \(p \over q\) where \(q\) is not equal to zero, is called a Rational Number. We represent the set of rational numbers by \(\mathbb{Q}\).
We note that the equation \(2x = 3\) , which is not solvable in \(\mathbb{Z}\) , is solvable in \(\mathbb{Q}\) . Thus, the solvability of equations increases in going from \(\mathbb{Z}\) to \(\mathbb{Q}\) .
✍Note: Every rational number can be reduced to the form \(\frac{x}{y}\), such that \(y \ne 0\) and \({\text{HCF}}(x,y) = 1\).
On reading this definition, one of the first questions that pop into our minds is, what are rational numbers? Well, think of all the numbers you have read about till today, in fact, let’s list them out shall we?
Integers, Fractions, Decimals, Whole numbers, Natural numbers, yes before we go any further, all these numbers are rational numbers.
Why only a fraction? You may ask. We just listed a whole bunch of names of the numbers you have been introduced to, so then why limit ourselves to just fractions. Well, think about it;
Can \(5\) be written as a fraction? Of course it can! It’s \({5 \over 1}\) where the denominator is \(1\) (non-zero). However, we deliberately didn’t use decimals in our description, it’s because while all fractions can be represented as decimals, not all decimals have a fractional representation. The square root of two and Pi are just two such examples.
Rational numbers, like integers, fractions and decimals before it, also incorporates the four basic operations. The way these operations work is identical to what were followed for fractions. Click on the following link to get a more in-depth look at the operations using rational numbers:
Challenge 1:
\[4,{\text{ }}0,{\text{ }} - 5{\text{, }}\sqrt {10} ,{\text{ }}\sqrt {1 + \frac{5}{4}} ,{\text{ }}\pi ,{\text{ }}\frac{\pi }{{\pi + 2\pi }}\]
Which of the above will represent a rational number?
⚡Tip: Try reducing to the form \(\frac{x}{y}\), such that \(y \ne 0\).
How do I understand?
The Foundational Nature of Rational Numbers
Essentially, rational numbers represent the supergroup of all the different kinds of numbers that children have been exposed to thus far. It is very important that students develop a keen understanding of not only this topic, but also how this topic evolves and emerges from what has already been learnt.
How Do You Visualise Rational Numbers?
Remember the number line and how crucial a role it played in bridging the gap between numbers and their associated values and visualizing the same? Well, guess what, the notorious number line is going to be your best friend once again! Let’s take a look at what rational numbers look like on the number line!
The Rational Line:
Suppose that we plot all the rational numbers on a number line. Let us call the plot we get, the Rational Line. Obviously, the Rational Line contains all the integers, but it also contains infinitely many points between any two successive integers. To understand this better, let us do a thought experiment. Suppose that you are walking on the number line, with the purpose of locating the rational number closest to 1, and just larger than 1. As you move towards 1 from the right side, you keep examining the rational numbers you come across, in the hope of eventually locating a rational number which is just to the right of 1, very, very close to 1.
However, what is really going to happen is something like this: no matter how close a rational number you find to 1, there will still be infinitely many rational numbers lying between that number and 1. In your quest to locate the rational number closest to 1, you will fail. In the rational set \(\mathbb{Q}\), we cannot talk about the closest number to any given number.
In fact, between any two rational numbers, no matter how close they are, there will always exist infinitely many rational numbers. A simple proof is as follows. Suppose that the two rational numbers areand. A rational number lying between the two can be calculated as the average of the two numbers, or \({N_1}=\frac{{\left( {x + y} \right)}}{2}\). Now, you can find another rational number between and this number as their average:
\[{N_2} = \frac{{\left( {x + {N_1}} \right)}}{2} = \frac{{\left( {x + \frac{({x + y})}{2}} \right)}}{2}\]
Going further, you can find another rational number between \({x}\) and this number, and so on, and infinitum. This is a simple way of showing that between any two rational numbers, no matter how close, there will exist infinitely many rational numbers.
Is there any closest rational number to 1?
The answer is NO. There is no closest rational number to 1 (or for that matter any rational number).
There is a particularly useful thought experiment to aid your visualization process. Let’s call it the Zooming Thought Experiment.
Zooming Thought Experiment:
Draw a number line, and mark 1 on the number line. Next, mark a number which you think is very close to 1; for example, 1.00001:
Now, zoom into the interval between the two points. Try to intuitively feel that there will be infinitely many more rational numbers on this segment (in fact, one such number will be the mid-point of the two numbers). On the zoomed line segment, let us mark a rational number close to 1 by a blue marker. Now, we zoom onto the interval between 1 and the blue marker:
In this zoomed segment, you can take a rational point close to 1 again. Let us represent this new rational number by a green marker. Now, we zoom onto the interval between 1 and the green marker:
We can continue this process as long as we want to. The intervals we are considering will keep on shrinking, but we will always be able to find rational numbers in each interval, no matter how small it gets.
This means that we cannot talk about the closest rational number to 1, since no matter how close to the number 1 we get, there will still be infinitely many rational numbers between us and 1. The three figures above can be combined into the following unified picture:
✍Note: There are infinitely many rational numbers between any two distinct rational numbers.
Solved Example:
Example 1: Find three rational numbers between \(\frac{5}{7}\) and \(\frac{9}{11}\) by using the mean method.
Solution: In the mean method, we will find the midpoint of any two distinct rational numbers and that will lie between those rational numbers.
Now, let \(x\) be the midpoint of \(\frac{5}{7}\) and \(\frac{9}{11}\), which means that,
\[x = \frac{{\frac{5}{7} + \frac{9}{{11}}}}{2}\]
\[ \Rightarrow \boxed{x = \frac{{59}}{{77}}}\]
Now, let \(y\) be the midpoint of \(\frac{5}{7}\) and \(\frac{59}{77}\),
\[y = \frac{{\frac{5}{7} + \frac{{59}}{{77}}}}{2}\]
\[ \Rightarrow \boxed{y = \frac{{57}}{{77}}}\]
Now, let \(z\) be the midpoint of \(\frac{5}{7}\) and \(\frac{57}{77}\),
\[z = \frac{{\frac{5}{7} + \frac{{57}}{{77}}}}{2}\]
\[ \Rightarrow \boxed{z = \frac{8}{{11}}}\]
Therfore, the three rational number between \(\frac{5}{7}\) and \(\frac{9}{11}\) are \(\frac{8}{{11}},\frac{{57}}{{77}},\frac{{59}}{{77}}\).
✍Note: There are infinitely many more rational numbers lie between \(\frac{5}{7}\) and \(\frac{9}{11}\).
Challenge 2: Find five rational numbers between \(0\) and \(1\) by using the mean method.
⚡Tip: Use a similar approach as in example 1.
How do you Teach Rational Numbers?
Chunking: The secret behind mastering rational numbers lies in mastering the topics that lead up to it. So, when your child is practicing the application of rational numbers, feel free to throw in concepts belonging to integers, fractions and decimals. This builds your child’s ability to quickly recall previous concepts and also, allows them to appreciate the interconnectedness of mathematics as a subject.
Flash Cards: Flash cards are excellent learning tools. They keep you on your toes and can be pulled out at a moment’s notice for a quick dose of revision. Repeated practice and use of such flash cards keep concepts fresh in the mind of your child and ensure long term retention.
