# Properties of subtraction and division

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?

• See Addition as well as Multiplication for basic def and properties. Sep 7, 2020 at 12:30
• 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. Sep 7, 2020 at 12:31
• But from a didactical point of view, it is probably better to introduce Division as a "basic" operations. Sep 7, 2020 at 12:33
• @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.
– Meuk
Sep 8, 2020 at 5:56

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