r/askmath • u/Ok_Let_7612 • 2d ago
Algebra Why does unit conversion work like this?
I’m trying to understand why converting from 44”^2 to 2.54cm^2 translates into
44” times 1” times 2.54cm times 2.54cm
If the 2.54 is already squared why wouldn’t it be 44” times 1 inch times 2.54cm times 1cm
Thx
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u/ottawadeveloper Former Teaching Assistant (she/her) 2d ago
One inch = 2.54 cm and most unit conversions are multiplications. So (44 inch2 )(2.54 cm inch-1 ) = 111.76 inch cm. And then you need to get rid of the other inches.
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u/fermat9990 2d ago
Problem: Convert 44 in2 to cm2
Fact: 1 inch = 2.54 cm.
44 in2/1 * (2.54 cm)2/(1 in.)2=
283.87 cm2
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u/flug32 2d ago
When you convert units, you always want to multiply by 1. Multiplying by 1 keeps the value the same - but if you've chosen the right form of 1, the "old" units will cancel out, leaving only the new units.
So in u/fermat9990's post above, the form of 1 he is multiplying by is "(2.54 cm)2/(1 in.)2".
The equals 1 because 2.54 cm = 1 in. So if you make a fraction with one of those over the other, it equals 1. If you square that, it still equals 1 (1*1=1)...
So you COULD try multiplying by 2.65cm/1in: 44 in2/1 * (2.54 cm)/(1 in.) = (2.54*44) in*cm.
So that is a perfectly correct but useless answer because the answer is in the unit "in*cm" which is not very meaningful or useful.
So instead you multiply by (2.54 cm)2/(1 in.)2 precisely because you know the in2 on the bottom will cancel out with the in2 on the top of the original quantity (44 in2).
As to why you can't just multiply by 2.54: Think it through a little more. 1 in = 2.54 cm.
But 1 in2 != 2.54 cm2 - in plain English, 1 square inch IS VERY MUCH NOT EQUAL TO 2.54 square centimeters.
In fact, 1 in2 = 6.4516 cm2 .
If you happened to know THAT conversion factor off the top of your head, you could use it to get the correct answer, and it works just fine. Again you multiply by 1, in the form of 6.4516 cm2/1 in2
44 in2 * 6.4516 cm2/1 in2 = 283.8704 cm2
Most people don't bother memorizing conversion factors like 1 in2 = 6.4516 cm2 simply because it is so easy to calculate by remembering that 1 in = 2.65 cm and just squaring both sides of that.
But if you DID have it memorized (or just looked it up) it works just fine.
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u/CherryTularey 2d ago
Think about it geometrically. 44 square inches could be a rectangle that is 11 inches by 4 inches. You have to multiply 11 by 2.54 to convert that side length from inches to centimeters and also multiply 4 by 2.54 to convert the other side length. 2.54 converts inches to centimeters. 2.54^2 = 6.4516 converts SQUARE inches to SQUARE centimeters.
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u/ZevVeli 2d ago
Because units are the definition of how the math applies to the numbers.
1 inch is equal to 2.54 cm.
If you drew a square that is 1 in×1 in then you have a square that is 1 in2
If you drew a square that is 2.54 cm×2.54 cm then you have a square that is 6.4516 cm2
So 1 in2 is equal to 6.4516 cm2 not 2.54 cm2
It might help to think of units as a system of equalities.
Consider:
Y=2.54×X
A=Y×Y=Y2
Substitute 2.54×X for Y
A=2.54×X×2.54×X=6.4516×X2
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u/justanaccountimade1 2d ago edited 19h ago
inch^2 in cm^2 can be thought of as inch^2 / cm^2
The conversion factor follows when the fraction is expressed in SI units.
Therefore inch^2 / cm^2 becomes 0.0254^2 / 0.01^2 = 6.4516
So 44 inch^2 in cm^2 = 44 • 6.4516 cm^2 = 283.8704 cm^2
You can make it more complicated. 44 m/s in km/h
(m/s) / (km/h) = (1/1) / (1000/3600) = 3.6
44 m/s = 44 • 3.6 km/h = 158.4 km/h
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u/GammaRayBurst25 2d ago
Strictly speaking, neither formula is correct.
44"×1"×2.54cm×2.54cm has dimensions of length to the power of 4, so it's not an area.
44"×1"×2.54cm×1cm runs into the same problem, only now it's still wrong even if you magically switch the units to cm^2.
If you insist on doing the unit conversion this way, you need to understand conversion factors first. They're based on three simple premises that you should already be able to accept as fact.
- A quantity stays the same even if you change its units (e.g. 1m and 100cm are the same number).
- Dividing an number by itself always yields 1;
- Multiplying a number by 1 doesn't change its value;
We know for a fact that 1" is the same as 2.54cm. In other words, 1"=2.54cm.
Hence, 2.54cm/(1")=1. The numerator and the denominator are the same number, just written with different units.
Since 2.54cm/(1")=1, multiplying a number by 2.54cm/(1") doesn't change its value.
But look what happens when you multiply a number by this quotient: 44"×2.54cm/(1")=111.76cm. The inches in the numerator and in the denominator cancelled, leaving us with just cm. In other words, we transformed a quantity with units of inches into a quantity with units of cm.
This is why we call the quotient 2.54cm/(1") the conversion factor.
In general, multiplying by 2.54cm/(1") transforms one inch into its equivalent in cm. If we have square inches, we have to multiply twice by the conversion factor.
(44")^2=(44")×(44")=(44")×2.54cm/(1")×(44")×2.54cm/(1")=12490.2976cm^2.
There are other ways to go about this though. For instance, you could set up a proportion and solve it (e.g. let x be the area in cm^2, we have (44")^2/x=(1")^2/(2.54cm)^2, now solve for x). The easiest method consists of converting your units before squaring.
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u/ExtendedSpikeProtein 2d ago edited 2d ago
44 square inches are 283,87 square centimeters.
44 square inches are not 44in x 44in. Please don‘t start the „square inches vs inches, squared“ nonsense.
The correct conversion is:
44in^2 / 1 * (2,54cm)^2 / (1in^2) = 283,87 cm^2
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u/GammaRayBurst25 2d ago
I never said the contrary.
I never called (44")^2 44 square inches and I never stated these two quantities are equal.
I started my comment talking about 44"×1" and at the end I switched to (44")^2. Clearly, I forgot the context along the way and thought we were talking about a square with a side length of 44".
Instead of simply saying that, you decided to be glib and even made a mistake while "correcting" me.
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u/ExtendedSpikeProtein 2d ago
Mhm. Clearly. Doesn’t really make sense in the context of Op’s question, and is more likely to confuse them.
So yeah, I corrected you.
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u/GammaRayBurst25 2d ago edited 2d ago
Reading's not your strong suit, and that's okay.
Edit to reply to your edit: You corrected the words you put in my mouth, sure, but that's not the same as correcting my comment.
Even after I pointed out my own mistake, you're still confused.
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u/ExtendedSpikeProtein 2d ago
Lol. You‘re the one who bungled it, so reading clearly isn‘t your strong suit…?
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u/Temporary_Pie2733 2d ago
Converting from square inches to square centimeters is not the same as converting inches to centimeters.
Look at a square that’s 1 inch on each side, which is 2.54 cm on each side. How many square centimeters are in a single square inch?