The Plane of Existence

Division, a fourth mathematical elaboration of counting is used in experiments where the measurement at the end is a proportion between two other measurements. If you are creating hydrogen from water, twice as much water will yield twice as much hydrogen. The predicted result is a proportion between the amount of water and the amount of hydrogen. The experiment involves doing the same procedure several times with different amounts of water. The fraction of the amount of hydrogen over the amount of water will be the same each time, though the numerator an denominator will vary. The proportion is th prediction. In this experiment doing the same thing a number of times is part of a single experiment. 

 

Whereas multiplication is a procedure that can be applied to making rectangles of tokens and counting their number, division is a process of arranging a number of tokens in a rectangle with a side of a certain size and announcing what the size of the other side is. The well known division sign “÷”, an instruction for doing this operation. 6 ÷ 2 = 3 can also be written 6 / 2 = 3, that is, as a fraction. But 6/2 = 12/4 = 18/6. A fraction can be written with many different numbers but have the same "value." A fraction can be used as a proportion. The final measurement, the proportion, is repeated in each case. If an experiment produces a final measurement that is a proportion you can double the recipe and get the same result.

 

An experiment can have the measurement of a proportion as its purpose. Equations used as instructions for experiments can frequently be written in the form A/B = 1. A and B are expressions for mathematical operations done with measurements. The numbers A and B can be different in different copies of the experiment, but the proportion is not. The experiment might predict a proportion between temperature, measured by the expansion of a coil of metal, and the height of a column of mercury. 

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There are many more mathematical operations, and we continuously devise new ones. It should not be too difficult to follow the instructions for each one and combine them according to rules learned, usually in grade school.