Why Do We Add Zeroes When Calculating the LCM and HCF of Decimals?

[Fiona Rozario]

Why do we make the number of decimal places equal (by adding zeroes) while calculating the LCM and HCF of decimals? I need to understand this because if I convert the decimals to fractions (without adding zeroes) and calculate the LCM of the fractions, the answers differ.
For eg: LCM of 1.2, 0.60, 0.144 is 3.6 if I make the number of decimal places equal and make them 1200, 600 and 144. But the LCM is 72 if I find the LCM of 12/10, 60/100 and 144/1000.

 

[Dr. Courtney]

What are LCM and HCF?

 

[HallsofIvy]

Before anyone can answer your question, you will have to tell us what you are talking about. The terms LCM (Least Common Multiple) and HCF (Highest Common Factor) apply only to integers. I have never seen them applied to decimal fractions. Perhaps whatever peculiar application this is is multiplying each decimal fraction by a power of 10 in order to have integers so that the LCM and HCF of those integers. But I would like to see what justification there is for such a method!

 

[Fiona Rozario]

 

[Fiona Rozario]

I can understand that by multiplying by a power of ten we are converting the decimals to integers and placing the decimal point back in place in the answer. However, if I find the LCM of the fractions 12/10 and 225/10 using the formula (LCM of numerators/HCF of denominators), I am still find LCM and HCF of integers. But the answer differs (quite obviously). But does that mean this method is wrong (using fractions)?

 

[nasu]

How is it different with fractions?

 

[pbuk]

Before anyone can answer your question, you will have to tell us what you are talking about. The terms LCM (Least Common Multiple) and HCF (Highest Common Factor) apply only to integers. I have never seen them applied to decimal fractions. Perhaps whatever peculiar application this is is multiplying each decimal fraction by a power of 10 in order to have integers so that the LCM and HCF of those integers. But I would like to see what justification there is for such a method!

Actually they can be defined for any commutative ring, however it is trivial that neither HCF nor LCM exist for the rationals so applying these terms to decimal fractions is meaningless.

The method in the posted image (which appears to define q = HCF(a, b) as the highest q for which integers (n, m) exist such that a = nq; b = mq) is novel, but I'm not sure it leads anywhere?

 

[Fiona Rozario]

In the example in the image, the LCM and HCF by that method and by converting to fractions (12/10 and 225/10) is the same. If I use the fractions 120/100 and 225/10, the LCM is 180. The example in my original query has been verified and the difference in answers is true.

Nonetheless, if there were to be a situation where we were required to find the LCM of decimals, I could use one method and someone else would use another method and both end up with different answers without being logically wrong in their respective methods. But that shouldn't be the case. What could be the justification?

 

[pbuk]

Nonetheless, if there were to be a situation where we were required to find the LCM of decimals, I could use one method and someone else would use another method and both end up with different answers without being logically wrong in their respective methods

That's the problem - the two methods you describe generally arrive at different results because they are different methods. Furthermore the concept of a Lowest Common Multiple is generally understood as applying only to integers; if you extend the concept to the rationals then you have the problem that any rational is a rational multiple of any other and so there is no LCM.

Who has told you that the result of either of the methods you describe can be called a Lowest Common Multiple?