In dimensional analysis what is a fraction that always equals one called?

Answer 1

I'm not exactly sure what is being asked, but it does at least allow us to explore reciprocals.

First, any fraction where the numerator is equal to the denominator will equate to one: 2/2, 3/3, 49/49, and 131/131 all reduce to 1/1 = 1. Similarly, x/x, y/y, and (a+b)/(a+b) all reduce to 1/1 = 1.

Second, the same holds true for dimensional analysis: meter/meter, foot/foot, ohm/ohm, and watt/watt all equate to one.

Third, if you multiply a fraction by its reciprocal, it will equate to exactly one. For example, if you multiply 3/4 by 4/3, it will equate to one. If you multiply 2/7 by 7/2, it too will equate to one.

Similarly, if you multiply a ratio of units -- that is, a complex unit expressed as a fraction, such as miles per hour (mi/hr) or meters per second (m/s) -- by its reciprocal, it too will equate to one. For example, if you multiply mi/hr by hr/mi, it will equate to one.

Taking it a step further, let's say you are given the following ratio: 16 cups per gallon (c/gal). The reciprocal of that quantity is 1/16 gal/c, which equals 0.0625 gal/c. In other words, if you multiply 16 c/gal by 0.0625 gal/c, it will equate to exactly one.