What Is Positive Times Negative

There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.

Mathematical consistency and patterns

Try solving each of these bug, paying attention to the previous set of bug as yous do then. Wait for patterns to brand solving the problems easier.

3 × three = ?
3 × 2 = ?
3 × ane = ?
3 × 0 = ?
3 × -1 = ?
3 × -2 = ?
three × -iii = ?
two × -3 = ?
ane × -3 = ?
0 × -iii = ?
-1 × -3 = ?
-two × -iii = ?
-3 × -three = ?

The answers to these bug are below simply I really do recommend taking the fourth dimension to solve the bug higher up on your own first, so you get the sense of how students might recollect through this set of problems.

iii × three = 9
iii × 2 = vi
3 × 1 = 3
3 × 0 = 0

At this stage, many people volition observe the answers are 3 smaller each fourth dimension and the number being multiplied by three is ane smaller each time, so they continue that pattern to answer the following questions.

3 × -one = -3
3 × -2 = -6
3 × -iii = -ix

Now, we decrease the start number in the pattern past 3 and one has to brand some deductions about what the reply should exist.

ii × -3 = -half-dozen
i × -iii = -3
0 × -iii = 0

One might now notice that the answers are going up by three each fourth dimension every bit nosotros increment the first number, then it is reasonable to go on this pattern.

-one × -3 = three
-2 × -three = vi
-three × -3 = 9

While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they take less cerebral capacity available to attain the task at hand (multiplying numbers together) and accomplish the boosted task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful.

Prerequisite knowledge: Ane has to know what these symbols mean, what is meant by finding ane number times some other, and how negative numbers piece of work in terms of counting down and subtraction.

Mathematical consistency and mathematical properties

Let's look at a problem that we tin can do in more than than one way, borrowed from the Khan Academy.

v × (3 + -iii) = ?

If we add the numbers inside the parenthesis first, then this is 5 times 0 which is 0, since 3 + -3 = 0.

five × (three + -3) = 0

Just what if we distribute five through both terms commencement?

five × 3 + 5 × -3 = ?

Since distributing the v across the addition does not change the value of the expression, nosotros know this is still equal to 0.

v × 3 + v × -3 = 0

Only this means that 5 × iii and 5 × -3 are opposite signs, so since 5 × three = 15, then 5 × -3 is -15. Let's wait at another example.

-5 × (iii + -3) = ?

We know that this is the aforementioned every bit -five times 0, and so this has a value of 0.

-5 × (3 + -three) = 0

Like to before, nosotros distribute -5 through both terms.

-v × 3 + -5 × -3 = ?

Once again, the distribution of terms does non change the value of the expression on the left-manus side of the equation, then the result is still 0.

-5 × three + -5 × -3 = 0

We know from earlier that -5 × 3 is -15 and then we tin can substitute that value for -5 × 3 in the left-mitt side of the equation.

-15 + -5 × -3 = 0

Therefore -15 and -5 × -3 are opposites since they add to 0, so -5 × -3 must be positive.

Goose egg in what we did for the two examples higher up is specific to the value of v × 3, and so we tin repeat this argument for every other multiplication fact nosotros want to derive, so these 2 ideas can be generalized.

Prerequisite noesis: 1 has to know what these symbols mean, what is meant past finding one number times another, how the distributive property works, and how negative numbers can be defined every bit the opposites of positive numbers.

Representation on a number line

Imagine we represent multiplication as jumps on a number line.

For 3 × 3, we draw three groups of iii moving to the right. Both the number of groups and the direction of each group are to the correct.

Simply what about iii × -3? Now we have iii groups of the number still, but the number is negative.

If we find -3 × three, the size and management of the number we multiply are the same, but now we are finding -3 groups of that number. One mode to think of this is to think of taking iii groups of the number away. Another is to remember of -3 times a number every bit being a reflection of 3 times the same number.

So -iii × -3 is, therefore, a reflection of 3 × -3 across the number line.

In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers every bit being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.

Context

Karen Lew has this analogy.

Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, nosotros are getting less negative.

This illustration between multiplication and addition and subtraction helps students nicely connect the two concepts.

Joseph Rourke shared this context.

A gambler loses $10 per twenty-four hours. How much more than money did they have 5 days agone?

Here, the loss per twenty-four hour period is i negative and going backwards in time is another.

@M_Teacher_w_T shared this analogy:

"An enemy of my enemy is my friend."

This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers.

Prerequisite knowledge: All contexts that build new agreement require students to understand the pieces of the context adequately well, and then it is specially important to probe how students empathise an idea when information technology is presented contextually.

Algebraic proof from start principles

From Dr. Alex Eustis, we have this algebraic proof that a negative times a negative is a positive.

First, he states a set of axioms that utilise to any ring with unity. A ring is basically a number system with two operations. Each performance is closed, which means that using these operations (such as add-on and multiplication on the existent numbers) leads to some other number within the number system. Each operation as well has an identity element or an element that does non modify another element in the system when practical to information technology. For example, under addition, 0 is the additive identity. Nether multiplication, 1 is the multiplicative identity. The full prepare of axioms required is below.

Axiom ane: a + b = b + a	(Additive commutivity)
Precept 2: (a + b) + c = a + (b + c)	(Additive associativity)
Axiom 3: 0 + a = a	(Condiment identity)
Precept four: There exists −a satisfying a + (−a) = 0	(Additive inverse)
Axiom 5: 1 × a = a × 1 = a	(Multiplicative identity)
Axiom 6: (a × b) × c = a × (b × c)	(Multiplicative associativity)
Axiom 7: a × (b + c) = a × b + a × c	(Left multiplicative distribution)
Precept viii: (b + c) × a = b × a + c × a	(Correct multiplication distribution)

From these axioms, we can evidence that a negative times a negative is a positive. I'll reproduce Dr. Eustis'south proof below and include the reference to the axioms used. First, nosotros bear witness that a = −(−a).

Corrolary 1

a = a + 0	(Precept 3 and Axiom 1)
a = a + (−a + −(−a))	(Axiom iv applied to −a )
a = (a + (−a)) + (−(−a))	(Axiom 2 – the associative property)
a = 0 + (−(−a))	(Axiom 4)
a = −(−a)	(Axiom 3)

And then now we know that if we introduce negative numbers a is equal to −(−?).

Corrolary 2

0 = a + (−a)	(Axiom 4)
0 = (0 + 1) × a + (−a)	(Axiom 3 and Precept 5)
0 = 0 × a + 1 × a + (−a)	(Axiom viii)
0 = 0 × a + (a + (−a))	(Axiom 5 and Axiom 2)
0 = 0 × a + 0	(Axiom 4)
0 = 0 × a	(Axiom 3 and Precept 1)

Proving that 0 = 0 × a is the kind of painfully obvious idea that hardly requires proof only it establishes a relationship between multiplication and the additive identity in the existent numbers, which is not even so included in the axioms above.

Side by side, we prove that (−1) × a = −a.

Corrollary iii

−a = −a + 0 × a	(Corrolary 2 and Precept 3)
−a = −a + (1 + (−1)) × a	(Axiom 4)
−a = −a + 1 × a + (−one) × a	(Axiom eight)
−a = (−a + a) + (−ane) × a	(Axiom 5 and Axiom 2)
−a = 0 + (−1) × a	(Axiom 4)
−a = 0 + (−1) × a	(Precept 3)

Now, finally, we can evidence that (−a) × (−b) = ab.

(−a) × (−b) = (a × (−1)) × (−b)	(Corrolary 3)
(−a) × (−b) = a × ((−ane) × (−b))	(Axiom 6)
(−a) × (−b) = a × (−(−b))	(Corrolary three)
(−a) × (−b) = a × b	(Corrolary 1)

This last "proof" though is unlikely to justify that a negative times a negative is a positive for any students though. Information technology's the kind of thing which is a required level of justification for a mathematician interested in rigorous proof who would likely consider the other justifications "patterning" and non sufficient.

A critical idea of proof though is that the intended audience of a proof is left convinced that an idea is true, and so I posit that the algebraic "proof" presented here is no proof at all for almost everyone.

Prerequisite knowledge: While I went through and added the justification for each stride of the proof that was missing, I needed a fair fleck of fluency with the original set of axioms. I also needed to not lose sight of the overall goal and to exist able to recognize the structure of each function of the argument and match that construction to the axioms.

A simpler algebraic proof

This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic.

a + (−a) = 0
a × b + (−a) × b = 0 × b
ab + (−ab) = 0

From this, we can evidence that ab and –ab have contrary signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is as well negative.

Similarly, we can bear witness that a negative times a negative is a positive.

a + (−a) = 0
a × (−b) + (−a) × (−b) = 0 × (−b)
−ab + (−a) × (−b) = 0

Since we know that −ab is negative, and the sum of these ii terms is 0, therefore (−a) × (−b) is positive.

Prerequisite knowledge: The prerequisite knowledge for this proof is much less than the other one, but it does assume a off-white bit of fluency with manipulation of algebraic structures.

Conclusion:

Given that the goal of an argument that something is truthful is to leave the other person convinced of the truth of the argument, whenever anyone uses whatever justification, representation, or proof, information technology behooves one to check that one's audience is left convinced.