In this post, I will try to address some misconceptions about numbers at the secondary level. In India, this includes classes 9 and 10. I will also make an attempt to provide reasons why these misconceptions occur and what we need to do to remove them.
Misconception #1:Rational numbers are ONLY those numbers that are written in the p/q form.
Explain students the definition of rational numbers. Then, ask them to write five rational numbers. Go around and see how many of them included integers in their list.
In most books in India, we continue to use the term 'natural numbers', 'whole numbers', 'integers' even after introducing rational numbers. This practice creates a differentiation in the mind of students between these numbers. So, for students 4 is an integer (or a natural number or a whole number) but 4/3 is a rational number. And, he feels that only if 4 is written as 4/1 then it is considered a rational number.
To remove this misconception, it is important to refer to all numbers as rational numbers once this has been introduced.
Question: Sumit ate 2/5 of a pizza, Rani ate 3/5 of a pizza and Bunti ate 1/2 of a pizza. If we added up the parts of pizza that two of these friends ate, in which case, the final answer will NOT be a rational number:
a) Bunti and Rani
b) Bunti and Sumit
c) Sumit and Rani
d) None of the above.
Misconception #2:There are 2 numbers that lie between 1 and 4.
Ask students to pick any number less than 100 and write it down on a paper. Collect these papers. See how many of them did not pick a positive integer.
Even after learning about rational numbers and knowing that there are infinite "numbers" between any two distinct numbers, most students think of numbers as counting numbers. Why does this happen?
To a large extent this is related to the first misconception. They feel that they need to pick 75/2 only if there were asked to pick "rational numbers". And that a 'number' still means a natural number. So, in the real sense, mathematically, these students have not really 'expanded' the number system in their heads.
To remove this misconception, it is important that students get to recall and apply these concepts as often as possible. As compared to integers, students use rational numbers in a very limited way - and that too only in a math class. Some students do not even consider decimal numbers as rational. However, worksheets designed on these concepts and inclusion of such questions in the exams can help them think more.
Misconception #3:There are two square roots of a positive number.
So, what exactly is the square root of 25? Is it 5, -5 or both?
Many students (and a lot of teachers) develop the misconception that square root of 25 is 5 and -5 both. The roots of this misconception lie in the concept of squaring a number and the fact that we are taught that finding the square root is the reverse process of finding the square. So, if the square of 5 and - 5 is 25, then should the square root of 25 not be 5 and -5 both? And, is this not why we write the sign +/- in front of squareroot sign.
So, here is the explanation.
The square root of 25 is 5. The square root of A is defined as the non-negative number B whose square is A.
But, its true that there is also another number -B whose square is also A. So, in situations, where the problem demands us to use both these numbers, (such as quadratic formula), we write the +/- sign to indicate that we need to use both the positive square root and its negative.
And, this almost always happen when we are trying to solve equations of the form x^2 = A. In such situations, there are 2 numbers that satisfy this equation. One of these numbers is B and the other is -B. So, we write x = +/-(B).
But, if we were asked to determine the square root of A, the answer will be B.
Misconception #4:Every number has a unique representation.
Every integer has a unique representation. So, a 4 can be written only in 1 way unless we write it as a result of a binary operation between 2 numbers (1.5 + 2.5).
However, rational numbers have two representation forms:
a) the p/q form, and
b) the decimal form
And, within the p/q form, there are several equivalent representations.
Thus, 1/2, 0.5, 151/302 are all same numbers.
BTW: 0.999.... and 1 is also the same number.
Misconception #5:0.999... and 1 are two different numbers.
I have no idea how many times I have encountered this misconception is some of the smartest people I know. This is one of those things that we just can't believe. How can these be the same numbers? It seems very obvious that no matter how far you write 0.999...., it will still be less than 1. Well, yes.
If after writing 10 million 9's after the decimal point, you STOP and remove the '...' then the number you wrote is definitely less than 1. But, the moment you put the '...' back, it becomes equal to 1.
There are several ways this can be proved. But the beauty is that we do not trust these proofs. So here are two arguments that I have used in the past.
Argument of contradiction: If suppose these were not equal, then by the denseness property of rational numbers, we should be able to locate infinite rational numbers between them. But, can we?
Direct proof: This is usually shown in grade 9 textbook in India.
Let x = 0.999...
Then, 10x = 9.99...
Subtracting these two equations, we get (aahhh...) 9x = 9. This means, x = 1. But, our x was 0.999..... Thus, it is proved that 0.9999.... = 1.
As always, all comments, queries, and expansions to this list are most welcome :-)
About the Author
Dr. Atul Nischal is a mathematician with 30 years of teaching experience of school and college math courses. He obtained his PhD in mathematics from Tulane University, New Orleans, USA. He is currently serving as the Managing Director of ECPL, an international award winning math education company based out of Delhi, India. In this position, he is involved in designing and developing online courses for students and professional development of teachers. He can be contacted at firstname.lastname@example.org