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In many textbooks, the unit factor method for converting units is described in this way:

In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa.

I know how to do it, i know how to calculate it, but i don't know why it works.

Mathematically 100 kPa / 1 bar isn't equal to 1. It doesn't make mathematical sense to consider it as 1.

It just doesn't feel right to me to consider it as 1 and hope it acts as 1 and then cancel things out.

The proof that multiplying by the conversion factor is the same as multiplying by 1 doesn't seems rigorous at all.

Karine Silva
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  • Given a quantity in bar, expressing the same quantity in kPa requires multiplying the number associated to the first quantity by 100. This is just a re-expression of the same quantity, so you have the same quantity even though it's represented by a different number. "Multiplying both sides by 100 kPa/1 bar" is just a shorthand way of saying this. – Ian Oct 03 '18 at 18:19
  • The real power of this shorthand arises when you apply it to quantities with composite units, like the gravitational constant $G$, because you can do a bunch of such multiplications in succession to derive a relationship between two composite units on the fly. – Ian Oct 03 '18 at 18:22
  • (Also, the ratio of two equal quantities with the same dimensions (not units, dimensions), really is just the dimensionless number 1.) – Ian Oct 03 '18 at 18:28
  • Hi! I do know that the shorthand does that, the problem is the argument that we can simply add it to the equation and it won't make any difference because it is equal 1, the thing is that it isn't mathematically equal to 1. – Karine Silva Oct 03 '18 at 19:01
  • It actually is, but it's a bit hard to see why because we're attached to units rather than dimensions. The point is that something like 1 meter is not the phrase "1 meter" literally, it is a certain amount of length which has meaning irrespective of the system of units we use to describe it. "100 cm" is exactly the same amount of length, so it represents exactly the same quantity. Thus "(1 meter)/(100 cm)" is a quantity divided by exactly the same quantity, which can only be the dimensionless $1$. It's like writing "x/y" when x=2 and y=2. – Ian Oct 03 '18 at 19:12
  • Another way to think about it: consider referential transparency. Imagine an object, call it X for brevity, with length 1 meter (maybe the original metal bar that was used to define the meter). Presumably, "(the length of X)/(the length of X)"=1; this is not just the same quantity but even the same way of describing the same quantity. But "the length of X" is 1 meter and by definition 100 centimeters simultaneously, so by substitution you get "(1 meter)/(100 cm)"=1. – Ian Oct 03 '18 at 19:16

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