My textbook clearly states after the lesson on Transforming Equations: Addition and Subtraction
"Notice that the subtraction property of equality is just a special case of the addition property, since subtracting the number c is the same as adding -c."
But, following the next lesson which is on Transforming Equations: Multiplication and Division there is no equivalent statement such as
The division property of equality is just a special case of the multiplication property, since dividing by the number c is the same as multiplying by the reciprocal of c.
Is division just a special case of multiplication, with the restriction of not having 0 in the denominator?
If so, why would they not include such a natural parallel statement?
Answer
I cannot speak for the authors, but my guess is that the relation between addition and subtraction is sometimes subtly different from that between multiplication and division, while sometimes the difference is more pronounced. For example, when dealing with real equations, division is simply multiplication by the inverse, except when dividing by $0$. However, in the case of $\mathbb{Z}$ and $\mathbb{R}[x]$, there are multiplicative inverses of only a few elements, whereas there are additive inverses of all of the elements.
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