why is L.C.M of rational numbers not posible?

Answer 1

Rational numbers have the special property that any rational number p/q is a rational multiple of any other rational number r/s.

p/q = (ps/rq) (r/s)

This property does not exist for integers, and hence we talk of least common multiple for integers, but not for rational numbers.

Answer 2

The reason that LCM in the rationals is not defined is the same reason that GCD and prime numbers in the rationals are not defined. Every non-zero number in the rationals is a unit. This means that for every number n/m that you can find in the rationals, there is also the number m/n in the rationals, and n/m*m/n=1.

Why does this make a difference?

Let's look at the LCM of a number dealing with integers:

the LCM(a,b,c,d)=m. This means that m is the smallest non-zero number such that a,b,c, and d divide m. Thus there exists integers u,v,x,y such that m=au=bv=cx=dy, and m is the smallest integer that you can do this with.

Now consider rational (non-zero) numbers s,t. Then 1/s, 1/t are also rational numbers. Thus 1/s*s+1/t*t=1, therefore s,t divide 1. But 1/(ns) and 1/(nt) are also rational numbers, and 1(ns)*s=1(nt)*t=1/n for any n>0, thus s,t divide 1/n for any n>0. This states that we can make 1/n as small as possible. Since the LCM is greater than zero, and 1/n-->0 as n-->infinity, the LCM is not defined.

I hope this makes since, if not please message me to clarify.


Ohh, and don't listen to EasternStar; not only is she rude, but she is also wrong. A rational number is a number that CAN be written as a ratio of integers. 1.2 can be written as 6/5 and is thus a rational number. I'm surprised you teacher didn't teach you that in 7th year when you were learning about LCM and rational numbers.

Answer 3

It's not so much impossible to extend the idea of LCM, or least common multiple, from integers to any rational numbers as a matter of how would you define it and what would you do with it. For example, maybe LCM(2.4,1.5)=2.4 because 2.4*1=2.4 and 1.5*1.6=2.4 or maybe LCM(2.4,1.5)=1.5 because 2.4*0.625=1.5 and 1.5*1=1.5
I just don't see a way of defining it for rational numbers that aren't integers that makes a lot of sense or helps solve a problem. The LCM defined for integers lets you find the least common denominator for adding or subtracting fractions in a simple form, without making the denominator itself a fraction. If you have a complex fraction, you'd want to simplify it, not come up with a complex fraction as the result.

As for the claim that decimal numbers aren't rational numbers anymore, it's just notation. For example, 1.2 = 12/10 It doesn't change the set the numbers belong to just because it's written differently.

Answer 4

I guess it's possible, like LCM( 2.4 , 1.5 ) = 12, but does it have a use?
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To formally define LCM for rationals:
Let A=(p/q), B=(r/s)
LCM(A,B)=AC=BD

where C and D are relatively prime numbers

so for above example, 2.4*5=1.5*8=12
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EasternStar, you are justifying your point by using a circular argument. Also, please understand that fractions can be converted into decimals, and vice versa.

I like to keep an open mind; however this seems to be progressing into an argument over definitions - these are usually futile.

Answer 5

are you 11 or 12? i think so, cuz in india, class 7th is taught all this.

anyway... LCM of rational nos. is not possible because of the same reason that LCM of fractions is not possible