Archive for September, 2013

Multiplication in Lilavati

Bhaskaracharya-II (1114 – 1193 CE) was one of the greatest Mathematicians that India has every produced. One of his treatises was ‘Lilavati‘ where he discusses various topics of mathematics. Below are a few links where we have made a presentation of different ways of multiplication discussed in Lilavati. Learn and Enjoy!

Multiplication #1

Multiplication #2

Multiplication #3

Multiplication #4


Who discovered Zero?

Understanding Mathematics

In schools, we see that many teachers and students try to focus on learning a lot of rules and formulae in mathematics without even trying to understand the ‘reasoning’ behind it. Since they manage to score good marks when they learn like this, they don’t see anything wrong in this way of learning. That is because, neither these students nor most of the school teachers are aware of the perils of this type of learning. Let me just try to explain it with an example.

Suppose there is a question on writing a given set of numbers in ascending/descending order. The kind of numbers they get in school for such a question are: 1/5, 3/7, -2/3, etc…which can be easily done by the way of finding LCM. Probably, the focus in school is to learn the ‘method’ of comparing fractions using LCM. But what if you get some different numbers like 13/18, 2/7, 14/19? If the students try to find the answers using LCM method, then they’ll take a lot of time in multiplying the denominators because they are bigger numbers as compared to the earlier question. And there’s a great probability of making an error in calculation too.

As a school student (or teacher), they might not be much bothered about the second question because they don’t get questions of this difficulty level in school exams. Here lies the problem!
The second type of question might be asked in an aptitude exam later in their life when they apply for a job interview or entrance exam for higher studies. At that time, they just know one method of solving the question….the rigorous LCM method…by which they end up using a lot of time (which they cannot afford in an aptitude test) and making errors. By this time, it’s almost too late and no one is going to teach them fraction-addition at this age.

Realizing this at an early age can be a blessing. While we are learning or teaching fractions in smaller classes, we can teach them alternative ways of comparing three fractions using logic. Let us now consider the second example of fractions again and compare them using logic.

A = 13/18,

B = 2/7,

C = 14/19

Comparing A and B using cross multiplication we get A>B. At this step, all we have to do is 13*7 and 18*2. Without even getting the results, we can look at those numbers and say that 13*7 will be greater than 18*2. So, we can immediately say A>B just by MERE OBSERVATION!

Now we can compare A and C. 18*14 > 13*19. Hence C>A. This is the ONLY CALCULATION one has to do in this question. (if a student is trained properly, they can say C>A by MERE OBSERVATION without actually finding the results 18*14 and 13*19 because average of 18 & 14 and 13 & 19 is 16. In such cases, the numbers nearer to the average, i.e.,18 and 14 in this case, will have a greater product than 13 & 19).
Thus we have got A>B and C>A, hence C>A>B!

In an aptitude test, the time given to solve this question might be 1 to 2 minutes depending on the type of exam. Now, many students (who can’t think…rather, who are not trained to think) might say that the second question will take more than 2 minutes to solve and hence the exam was difficult. But the fact is, they are not able to think logically and solve a problem…they just know one way of solving the problem. The world today does not want such people who cannot Think.

Thus, when we are teaching and learning such topics in school, we should be trying to find logical ways of solving along with conventional methods. This can happen when we shift our main focus of teaching/learning from Scoring Marks to UNDERSTANDING MATHEMATICS.