From this, we can show that ab and — ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative. Prerequisite knowledge : The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures.
Using the number line again, and considering just -1 as a multiplier and p as some positive number:. Next fun question: we tell students to reverse an inequality when multiplying both sides by a negative. March 15, — pm. I also like to show students that 9 x 9 which they know is 81 is the same as Expanding to give the first 3 terms using the distributive law or the grid or box method yields — 10 — 10 which gives Not a proof though!
March 25, — pm. If you add weight pos. If you remove weight neg. June 1, — pm. July 14, — pm. Yeah, this is a good explanation, mostly because it makes sense to me and to you! July 20, — am. Your email address will not be published. Notify me of followup comments via e-mail. You can also subscribe without commenting. This site uses Akismet to reduce spam. Learn how your comment data is processed.
Email address:. I have read the privacy policy. Mathematical consistency and patterns Try solving each of these problems, paying attention to the previous set of problems as you do so. We know that this is the same as -5 times 0, so this has a value of 0. Representation on a number line Imagine we represent multiplication as jumps on a number line.
Now we have 3 groups of the number still, but the number is negative. Context Karen Lew has this analogy. Algebraic proof from first principles From Dr. This number line is a simplified version, but you can draw them with every number included if you wish. The big advantage of a number line is that it is very easy to draw for yourself on the back of an envelope or a piece of scrap paper, and it is also quite hard to go wrong with the calculation.
As long as you are careful to count the number of places that you are moving, you will reach the correct answer. This time you start at and move 23 places to the right. You can see straight away that the answer is 6. If you subtract a negative number, the two negatives combine to make a positive. Instead, you can think of it as turning one of the negative signs upright, to cross over the other, and make a plus.
For clarity, you would never write two negative signs side by side without brackets. So if you are asked to subtract a negative number, it will always have brackets around it so that you can see that the use of two negative signs was intentional. Once you have the numerical answer, then you can apply a very simple rule to determine the sign of the answer:. As a side issue, this goes some way to explaining why you cannot have the square root of a negative number there is more about this in our page on Special Numbers and Concepts.
The square root is the number that is multiplied by itself to get the number. You cannot multiply a number by itself to get a negative number. To get a negative number, you need one negative and one positive number. The rule works the same way when you have more than two numbers to multiply or divide. An even number of negative numbers will give a positive answer.
An odd number of negative numbers will give a negative answer. Dividing positive and negative numbers. Practice: Dividing negative numbers. Why a negative times a negative is a positive. Why a negative times a negative makes sense. Multiplying negative numbers review. Dividing negative numbers review. Next lesson.
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