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first post ever on stack exchange in years of using it.

Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance theory, specifically the DuPont formula, we see that we can multiply by Sales/Sales to alter our result "intuitively" and come up with different results. Alternatively in some algebraic proofs we multiply by x/x and then rearrange. Etc.

What I can't seem to understand is why this works. We see this across mathematics, multiplying all sorts of equations by fractions equivalent to 1 so that they can be rearranged through cancelling out numerators and denominators. Why are we "allowed" to do this? Once we cancel we are left with residual denominators or numerators. Is there an easy way of visualizing how this is allowed?

someonesomewhere
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    Sounds like you are talking about various algebraic operations of real numbers. It is a postulate of those numbers that

    $$ 1 \cdot x = x \ \text{ for all real numbers } x$$

    This is also intuitively pleasing: if we have one of something, then we have the something, no more or no less.

     – Simon S Jun 15 '15 at 21:41
  • Multiplicative identity allows a multiplication to occur without changing the value. Because the value remains the same, they are equivalent mathematically. – Iceman Jun 15 '15 at 21:42
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    Are you comfortable with the rules for computing with fractions in general, like $(a/b)\cdot (c/d) = (ac)/(bd)$ for example? If so, this should be nothing but a special case. – Hans Lundmark Jun 15 '15 at 21:46
  • What do you mean by "fractional 1"? The fraction $\frac{1}{1}$ as factor? – mvw Jun 15 '15 at 21:47
  • good question, here's the simple answer. It rationalizes the equation into forms we can better study. – Zach466920 Jun 16 '15 at 02:35
  • Multiplying by $1$ doesn't change a number. That's why it's allowed. – anon Jul 19 '16 at 10:59

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Fractions have multiple forms.

Basically, what matters about fractions is the proportion they represent. Certainly a stadium exactly half full looks very different from a room with two chairs, only one occupied, but both are half full, and that proportion can be written as 5000/10000, as well as 1/2.

However, the form 1/2 is kinda special, in that 1 and 2 have no common divisors aside from 1; that is not the case for all the other forms, so, in a sense, 1/2 is the simpliest form.

When we simplify, we go from one form to a simplier form. When we multiply by a/a, we go from a form to a less simple form. All these operations are allowed because they don't change the proportion represented by the fraction in the slightest.

You can think of a scale, two plates with weights: there are ways to take or add weights to each scale without changing their position; that is to say, preserving how much one weighs in respect to the other.

byserpas
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