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QUESTION: Assuming I have 2 or more numbers and most of them are whole number but one has a few numbers right of the decimal. Assuming it isn't a nonterminating number how do I multiply all numbers up so they are all whole number in the simplist way?

ANSWER: Let's start simple: First, assume you have only one number, not a whole number, but you want to multiply it to make it a whole number.

For example, if you had 3.125, you could multiply by 1000. That is the same as moving the decimal to the right 3 places, giving you 3125.

However, 1000 is bigger than we need in this case. Multiplying by 8 gives you 33.

The way to do this is to write your decimal number as a fraction. You can ignore the part to the left of the decimal -- it is already a whole number, so you can multiply it by any other whole number without any trouble.

The 0.125 is the same as 125/1000. That's why I know 1000 will work. But I can reduce that fraction by 5 to get 25/200, and again by 5 to get 5/40, and one more time to get 1/8.

That 8 in the denominator is all I need to know.

Say I have a different number like 12.045. I can ignore the 12, and the 0.045 is the same as the fraction 45/1000, which can reduce to 9/200. The number I have to multiply by in this case is 200.


Now, if you have multiple numbers, you maybe have:

1
3.4
3.125
9.1
11


For each of these numbers, we can chop of the whole number, write as a fraction, reduce, and look for the denominator. We get the following:


1 -> already a whole number
3.4 -> 0.4 -> 4/10 -> 2/5 -> denominator 5
3.125 -> 8 as above
9.1 -> 0.1 -> 1/10 -> denominator 10
11 -> already a whole number


So the question is what to multiply by. If you use 5, it does not work, it only fixes the second number (not the third or fourth). If you use 8, it only fixes the third. If you use 10, it fixes the second and fourth, but not the third.

You could multiply by 5, then by 8, then by 10, but if you look again, clearly if we do the 10, the 5 is not required. And in fact, the 8 is not required either, if you did 10, you could follow up with 4 to fix the third one at that point.

What you need is the least common multiple of these numbers, which is 40 in this case.

---------- FOLLOW-UP ----------

QUESTION: My application is I want to determine the exact slope of various triangles on graphing paper, and one of the points for sure will be not a whole number when the Y axis line is set to 1. I found some tables online that show angles to y coordinate but don't know how much rounding is done and therefore whether many of those numbers terminate. I am trying to build shapes for 3D modeling. On thingiverse.com anyway I haven't been able to find an equalateral triangle pyramid. Just square based ones. How does one discover slopes without trusting their eyes on paper with protractor?

Answer
The slope of any line segment is the ratio of its height (measured vertically) to its length (measured horizontally. This is a very common question, so I can just point you to a website that has some pictures that I think will help: http://hotmath.com/hotmath_help/topics/slope.html

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.

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I am a PhD educated mathematician working in research at a major university.

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AMS

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Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

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BA mathematics & physics, PhD mathematics from a top 20 US school.

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Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

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In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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