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

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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.

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• 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.

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• 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.

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• 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.

Source(s): 2nd year engineering, not that this question has anything to do with engineering
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