I'm (re)learning math and was looking for the properties of the most basic math operations. For addition and multiplication these are very easy to find (e.g. https://www.aaamath.com/g74a_px1.htm and https://www.aaamath.com/mul74bx2.htm. I also tried to find this for subtraction and division and there are some list of properties but i also found this: https://www.quora.com/What-are-the-different-properties-of-subtraction-and-division here some people answered:

Addition and multiplication have properties, such as the commutative property. Division is actually multiplication of the inverse, and does not have its own properties. Subtraction is actually addition of a negative, and does not have its own properties.

and :

Subtraction is addition in a negative, so it doesn't have its own properties. Division is multiplication of the inverse, and it doesn't have its own properties.

Is there any truth in this? If so how can I use this when doing/solving subtraction/division problems?

  • $\begingroup$ See Addition as well as Multiplication for basic def and properties. $\endgroup$ Sep 7, 2020 at 12:30
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    $\begingroup$ With "Division is actually multiplication of the inverse, and does not have its own properties" the author of the quote simply means: due to the fact that division is defined in terms of multiplication, the properties of division are directly derived from those of multiplication. $\endgroup$ Sep 7, 2020 at 12:31
  • $\begingroup$ But from a didactical point of view, it is probably better to introduce Division as a "basic" operations. $\endgroup$ Sep 7, 2020 at 12:33
  • $\begingroup$ @MauroALLEGRANZA thanks for the reply! So for learning purposes i should look at division an subtraction as basic operations but theoretically they are just multiplication and addition. $\endgroup$
    – Meuk
    Sep 8, 2020 at 5:56

1 Answer 1


Similar to the addition operation, subtraction has its own special properties but they are not as widely stated explicitly. $$ 0 - x = -x.\tag{1} $$ $$ x - 0 = x. \tag{2} $$ $$ x - x = 0. \tag{3} $$ $$ x - y = -(y - x). \tag{4} $$ $$ (x - y) - (z - w) = (x - z) - (y - w). \tag{5} $$

There is a similar list in an answer to MSE question 1225445 "Abelian groups axioms with minus in place of plus".

  • $\begingroup$ Thanks for the list of properties. Do you have a source/website where these rules are explained further? $\endgroup$
    – Meuk
    Sep 8, 2020 at 5:58

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