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Math Help - Intuitive order of definitions in arithmetic

  1. #1
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    Intuitive order of definitions in arithmetic

    I have recently read through much of Edmund Landau's Foundations of Analysis but have wondered if there is a more intuitive order of defining fractions, division, negatives, subtraction, etc. Is there a book, website or even a video series anyone can recommend that presents such an order and clear motivation for all definitions?

    For instance, I wonder if it is more intuitive to define fractions first or the operation of division.
    Also, multiplying -1 by 3 makes sense: -1+-1+-1 but multiplying 3 by -1 doesn't seem as straightforward. I would like a clear motivation for defining it. Should the definition be motivated by the fact that multiplying a positive number by -1 is the same as subtracting twice the number: a * -1 = a - 2(a)
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    Senior Member roninpro's Avatar
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    I haven't seen much literature on this question, and it seems to me that people are reluctant to talk about it.

    To me, it seems that the idea behind 3\cdot -1 is that we want our numbers to commute under multiplication. So, we simply define 3\cdot -1=-1\cdot 3, which (according to your reasoning) we know how to compute.

    For division vs. fractions, I think that it would be more natural to have division first. As soon as you introduce that operation, you realise that there are many holes in your number system: if you try to compute 1\div 3, you cannot do it. This certainly would motivate the idea of fractions to fill up those holes.
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    For division vs. fractions, i think fractions come first by the way of the multiplicative inverse. You define \frac{1}{a} to be the multiplicative inverse of a, i.e.  a \cdot \frac{1}{a} = 1. Then you define  1 \div a as  \frac{1}{a}
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    I agree defining division first feels more natural and think it may be because you might not be able to draw a picture of a fraction without having divided something into equal groups/pieces.
    The motivation for fractions could be for measurements, but I'm trying to think if there's a need for multiplication of fractions or it's just a convenience. For instance 1 inch = 1/12 foot = 1/12 \cdot 1/3 yard but we could have stated:

    1 foot \div 12 = 1 yard \div 3   \div 12
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    Quote Originally Posted by lamp23 View Post
    a * -1 = a - 2(a)
    This seems like an unnatural and messy definition to me.


    Defining division before defining the rational numbers does not make sense (if you're defining the rational numbers in terms of the integers). Division cannot be defined for the integers (minus zero) because the integers are not a multiplicative group under multiplication (in particular not every nonzero integer has a multiplicative inverse).

    That said, there are many equivalent ways to define various number systems, and different textbooks in advanced mathematics define standard number systems in different ways.
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    Quote Originally Posted by lamp23 View Post
    I agree defining division first feels more natural and think it may be because you might not be able to draw a picture of a fraction without having divided something into equal groups/pieces.
    The motivation for fractions could be for measurements, but I'm trying to think if there's a need for multiplication of fractions or it's just a convenience. For instance 1 inch = 1/12 foot = 1/12 \cdot 1/3 yard but we could have stated:

    1 foot \div 12 = 1 yard \div 3   \div 12
    What about trying to find the area of a block of land that's \displaystyle 10\,\frac{1}{2}\,\textrm{ft} \times 9\,\frac{1}{4}\,\textrm{ft}?
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    Hmm, I'm certainly going to think about whether it's more intuitive to define fractions or division first.

    Anyway, is the point of making a definition that makes negative times negative positive just so that distances do not rely on which direction you make positive? Namely that |b-a| = |a-b| ?
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    Negative times negative is positive because it's the only definition that makes sense.

    You should have a fair knowledge of the distributive law, i.e. that \displaystyle (a + b)(c + d) = ac + ad + bc + bd. This is easily understood using a diagram...

    Say you wanted to evaluate \displaystyle 9 \times 9.

    Notice that \displaystyle 9 \times 9 = (10 - 1)\times (10 - 1) = 10 \times 10 + 10 \times (-1) + (-1) \times 10 + (-1) \times (-1) = 100 - 10 - 10 + (-1) \times (-1) by the distributive law.


    Now let's examine this using diagrams...

    \displaystyle 10 \times 10




    \displaystyle 10 \times 10 - 10 \times 1




    \displaystyle 10 \times 10 - 10 \times 1 - 1 \times 10

    .


    Notice that you have subtracted the corner unit one time too many, so that means to get \displaystyle 9 \times 9, you have to add that \displaystyle 1 unit back on.

    Therefore \displaystyle 10 \times 10 + 10 \times (-1) + (-1) \times 10 + (-1) \times (-1) = 100 - 10 - 10 + 1.

    Thus \displaystyle (-1) \times (-1) = + 1, and so negative times negative is positive.
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    So one reason is that we want to define -(-1) = 1 so that the distributive property works? I've been trying to come up with an example of why we want the distributive property to work:

    I owe you $9 but I only have a ten dollar bill on me so I give that to you and have you owe me one dollar.
    (your perspective): $10+-$1
    (my perspective): -($10+-$1) = -$10 + -(-$1) We know that you owe me 1 dollar so i should get plus one dollar after i pay u 10 so it definitely should be: -$10+$1
    Hmm, but I guess this doesn't really explain why we would want to even be able to distribute that negative if we could just simplify inside the parentheses.

    The other thing I thought of is so that way when we are simplifying expressions we don't have to worry if a variable is negative if we want to use the distributive property. For example if -1(x+1) shows up we don't have to worry if x is negative to make it -1*x+-1*1
    yet actually if we were trying to solve -(x+1) = -5 we could divide both sides by -1 first and obtain x+1=5
    but if were were trying to solve -(x+1) = 5 we would need either negative times negative or the distributive rule to solve it
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  10. #10
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    It seems like although we could come up with methods that didn't involve knowing how to multiply fractions or multiply a negative number times another negative number, that for consistency and therefore convenience we make the appropriate definitions so that we can keep using the same rules.
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  11. #11
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    If we consider the product of two numbers a,b to be the signed (oriented) area of the rectangle with side lengths a and b, then it makes geometric sense.
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    Well, we say it should be a product because it makes sense when it's whole numbers. I'm saying the reason why we come up with the method we do, multiplying fractions, is for consistency and therefore convenience. One could still solve for the area of a 1/2 foot by 1/2 foot square if they converted to a unit that was defined as 1/2 foot. Let's call it a halfafoot. Then you would have 1 halfafoot * 1 halfafoot = 1 halfafoot squared. But then if you wanted to compare this to something like a 3 feet by 3feet square, you would have to first convert feet to "halfafoots" so it certainly wouldn't be convenient as one would want a unit for every unit fraction. But in our current system we can compare in any units we desire.

    However, the initial question one might have when we define multiplication of fractions is why bother if we already have a method (e.g.:
    3 \div 3 = 1
    so why bother with:
    3 \times 1/3 = 1

    Yet, I guess the obvious answer is to be consistent with the formula for area and so that way we can also continue to use the commutative and distributive laws.
    Last edited by lamp23; February 14th 2011 at 08:51 PM.
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