DragginMath: The Distributive Property

You know that 3∗9 = 27. Observing this fact from a different angle, this also means that 3∗(4+5) = 27. Drag the + up into Blue Mode, then drop it on ∗. DragginMath just told you that 3∗(4+5) means the same thing as 3∗4+3∗5. If you evaluate this, the result is still 27. This works for any numbers: a(b+c) = ab+ac. Also, (b+c)a = ba+ca.

The full name for this is the Distributive Property of Multiplication Over Addition. It also works for multiplication over subtraction: a(b–c). Try it to see how that works. It also works for division under addition or subtraction: (a+b)÷c, but not for division over addition or subtraction: a÷(b+c). Try both to verify that the first one does something, but the second one does nothing. Is there a Distributive Property of Addition Over Multiplication? Enter a+(b∗c), then drag ∗ up onto + to find out. DragginMath doesn’t think so, and it is easy to find a counterexample with actual numbers.

Distribution can be invoked in two different ways: general and specific. The general way assumes you want operands distributed as far as they can be. The specific way assumes you want them distributed only to specific places you will indicate.

To see general distribution, enter a(b+c+(d+ef)). Drag the topmost + onto the ∗ attached to a. DragginMath looks under that topmost + for any other addition or subtraction operators, all the way down all branches of the tree, and distributes to their operands. If it encounters any other kinds of operators on its way down the branches of the tree, it stops. With general distribution, DragginMath does all of the work, finding all distributions that can correctly be done in that subtree of operators. General distribution happens when you drag up from the topmost + or – under ∗ or ÷.

To see specific distribution, enter a(b+c+(d+ef)). Drag the + attached to b (not b itself) onto the ∗ attached to a. In this case, you have told DragginMath where to stop distributing. It will distribute down to that node, but no farther. To do this correctly, it may have to distribute to side branches also, but not inside them. For example, notice that the distribution of a did not get inside the (d+ef) branch for this specific distribution, but it did for the general distribution.

General distribution is usually what you want. But sometimes it is too much. In those cases, specific distribution can do exactly what you want, but it sometimes takes several steps to do it.

DragginMath can do multi-factor distributions. For example, enter ab(c+d–e)÷f. Look carefully at the structure of that tree. Drag – up onto ÷. Now look carefully at the structure of the result.

This is the Distributive Property of Multiplication (or Division) over (or under) Addition (or Subtraction). There are other kinds of distributive operations, but they aren’t usually called that. Many go under the general heading of Rules of Exponents. Others seem to have no traditional name. Whether they have names or not, DragginMath can do both general and specific distribution for them. The drag-and-drop invocations follow the same patterns. Some can do multi-factor distributions, while others cannot. For those that cannot now, some might in the future, while others won’t ever because the math doesn’t work that way. In any case, if you want to know if DragginMath can do something, think up an example and try it.