A conversion factor is the same quantity in two units, so it equals one.
Conversion factor A ratio equal to one, used to change units without changing a quantity.
Step 1: Something to Notice
Watch first. The explanation comes later.
A spacecraft was lost because one team worked in pound-seconds and another in newton-seconds. Every number involved was correct.
What kind of check would have caught it?
Step 2: Find Out
Isolate the instrument and note how many figures a reported value may honestly carry.
Step 1 — Predict
Why does multiplying by a conversion factor not change a quantity?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
Put the unwanted unit on the bottom
The cancellation decides for you whether that was multiplying or dividing.
Chains cancel step by step
55 mph becomes 24.6 m s⁻¹ through two factors, with miles and hours canceling.
Derived units are handled algebraically
g cm⁻³ times cm³ gives grams, with nothing else to think about.
A dimensional check is cheap and partial
It catches wrong units and misses wrong dimensionless factors.
A conversion factor is a way of writing one
Since 1000 g equals 1 kg, the ratio between them equals one. Multiplying by one never changes a quantity, which is why unit conversion is always legitimate.
Units diagnose the setup
If the units do not cancel to the required answer, the arrangement is wrong before any arithmetic is done. Tracking them is a check on the reasoning rather than bookkeeping.
Step 4: A Common Mistake
Lots of people think
“You multiply or divide by a conversion factor depending on which direction the conversion goes, and you have to remember which.”
Step 5: Why Dimensional Analysis Can Find a Formula
The same idea somewhere new.
If a quantity depends on only a few variables, requiring the dimensions to balance can determine the form of the relationship up to a dimensionless constant. G. I. Taylor famously estimated the yield of the first atomic test from published photographs this way, reasoning that the blast radius could depend only on energy, air density and time, and finding the combination that gives a length. The technique will not supply the constant, but getting a formula's structure from nothing but units is a genuinely surprising amount of physics for the effort — and it is why dimensional reasoning is a research tool and not only a checking device.
Step 6: Think It Through
Practice makes it stick.
Which Way Up
Problem 1 of 2
How do you know whether to multiply or divide by a conversion factor?
A Missing Factor of Two
Problem 2 of 2
An equation has correct units and is missing a factor of 2. Will a dimensional check catch it?
Units That Cancel
1 of 5
What is a conversion factor equal to?
2 of 5
Where should the unwanted unit go?
3 of 5
55 mph in meters per second is about:
4 of 5
What must be true of every term in a valid equation?
5 of 5
What can dimensional analysis sometimes determine?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Do you have to remember whether to multiply or divide by a conversion factor?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Explain why dimensional analysis works and how it is used to check an equation.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- A conversion factor equals one, so it changes units and not quantities.
- Putting the unwanted unit on the bottom removes any choice of direction.
- Every term in a valid equation carries the same dimensions, which makes checking possible.
- A dimensional check is cheap and cannot catch a wrong dimensionless factor.