Axioms – Ratios

In Cecilia Hamm’s “Making Sense of Negative Numbers,” she states, “Arnauld (1612-1694) claimed that the basic principle of multiplication is that the ratio of unity to one factor is equal to the ratio of the second factor. i.e., given the product x , then x or x .” This is known as the axiom of ratios.

Classical mathematics sign rules state that multiplying two negatives yields a positive. Arnauld believed that this rule violated the principle that “the ratio of unity to one factor is equal to the ratio of the second factor.” For example: Given = -4 and = -5:

For 1/, it can be expressed as 1:-4 = -5:20. Here, “:” represents the ratio, which is the relative size of two numbers. This illustrates the problem: the ratio of the larger number 1 to the smaller number -4 is the same as the ratio of the smaller number -5 to the larger number 20. In other words, “a larger number is to a smaller number (1 to -4) as a smaller number is to a larger number (-5 to 20).”

For 1/, it can be expressed as 1:-5 = -4:20. Again, this shows the problem that the ratio of the larger number 1 to the smaller number -5 is the same as the ratio of the smaller number -4 to the larger number 20.

However, in Wave Numbers, only the comparison of Opposite Values’ magnitudes is possible. Wave Numbers use the absolute values of Opposite Values for this purpose, as follows.

R1 and R2

Given = 4^v and = 5^v:

• For 1^{^}/, |1^{^}|:|4^v| = |5^v|:|20^{^}| => 1:4 = 5:20.
  • Both ratios are of a smaller Counter to a bigger Counter. Therefore a breach of the basic principle of multiplication does not happen.  

R3 and higher:            

Given a = 4i^v and b = 5j^v and ab = 20^{^} :

• For 1^{^}/a, |1^{^}|:|4i^v| = |5j^v|:|20^{^}| => 1:4 = 5:20.
  • Both ratios are of a smaller Counter to a bigger Counter. Therefore a breach of the basic principle of multiplication does not happen.

Axiom of Ratios Conclusion

As shown, the Wave Number system supports the basic principle of multiplication stated by Arnauld. This is because the ratio of unity to one factor is equal to the ratio of the second factor.

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