DIMENSIONAL ANALYSIS
Here is a diagram to help convert between different units of length:
miles ..............
> km
×5280
×1000
∨
∨
feet .................
>m
×12
∨
inches
×100
×2.54
∨
> cm
To go opposite the direction of an arrow, divide by the given number. Do
you see what we’re actually doing? We’re just multiplying by 1 each time!
We never change the quantity we have: every move we make, we get an
equal quantity in different units by multiplying by 1 in a special form. As
Charlie explained, we can write 1 in many different ways:
1=
1 km
5280 feet
1 foot
2.54 cm
1 km
=
=
=
=
1 km
1000m
1 mile
12 inches
1 inch
Here is the same diagram with units:
miles ........................
> km
feet
× 5280
1 mile
× 1000m
1 km
× 121 inches
foot
cm
× 100
1 m
∨
∨
feet ...........................
>m
∨
inches
cm
× 2.54
1 inch
∨
> cm
Why, then, would we divide by 5280 to go from feet to miles? We are just
1 mile
multiplying by 1 = 5280
feet !
Question: How many meters is 3 feet?
1
2
DIMENSIONAL ANALYSIS
Solution. Start at feet. Go down, go over, then go up. Algebraically,
3 feet = 3 feet ×1 × 1 × 1
12 inches 2.54 cm
1m
= 3 feet ×
×
×
1 foot
1 inch
100 cm
12 inches 2.54 cm
3 feet
×
×
×1 m
=
1 foot
1 inch
100 cm
3 × 12 × 2.54
=
m
100
' 0.91 m