A Negative Times A Positive

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Multiplying Negatives

When We Multiply:

		Example
×	two positives make a positive:	3 × two = 6

×	two negatives make a positive:	(−three) × (−2) = 6

×	a negative and a positive brand a negative:	(−3) × two = −6

×	a positive and a negative make a negative:	iii × (−2) = −six

Aye indeed, two negatives make a positive, and we will explain why, with examples!

Signs

Let's talk nearly signs.

"+" is the positive sign, "−" is the negative sign.

When a number has no sign it usually means that it is positive.

Example: 5 is really +5

And we can put () around the numbers to avoid confusion.

Case: three × −ii can exist written every bit (+3) × (−two)

Two Signs: The Rules

multiply like signs

"Two like signs make a positive sign,
2 unlike signs make a negative sign"

Example: (−ii) × (+5)

The signs are − and + (a negative sign and a positive sign), so they are different signs (they are dissimilar to each other)

And so the issue must be negative:

(−two) × (+five) = −10

Example: (−4) × (−3)

The signs are − and − (they are both negative signs), and then they are like signs (similar each other)

So the issue must be positive:

(−4) × (−3) = +12

Why does multiplying two negative numbers brand a positive?

Well, first there is the "common sense" explanation:

dog does not want brocolli

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Practise not eat!" I am maxim the opposite (negative).

Now if I say "Do NOT non consume!", I am saying I don't want you to starve, so I am back to proverb "Consume!" (positive).

So, two negatives make a positive, and if that satisfies y'all, and then you don't need to read any more.

Direction

It is all most direction. Recall the Number Line?

Well here we take Baby Steven taking his first steps. He takes 2 paces at a time, and does this 3 times, then he moves 2 steps 10 iii = 6 steps frontward:

multiply 2 x 3

Now, Infant Steven tin can also step backwards (he is a clever petty guy). His Dad puts him dorsum at the start and then Steven steps backwards ii steps, and does this three times:

multiply -2 x 3

Once once more Steven's Dad puts him back at the start, but facing the other way. Steven takes ii steps frontward (for him!) but he is heading in the negative direction. He does this three times:

multiply 2 x -3

Dorsum at the start again (thanks Dad!), even so facing in the negative direction, he tries his backwards walking, once once again taking ii steps at a fourth dimension, and he does this three times:

multiply -2 x -3

So, by walking backwards, while facing in the negative management, he moves in the positive management.

Try it yourself! Try walking forwards and backwards, so over again simply facing the other management.

Play With It

But perchance y'all would like to run across information technology in action? Use the sliders below:

numbers/images/number-line-mult.js

More than Examples

Example: Money

Sam gives you 3 $ten notes:		+3 × +10 = you gain $xxx
Sam gives y'all three $ten debts:		+3 × −10 = you lose $30
Sam takes three $10 notes from you:		−3 × +10 = you lot lose $30
Sam takes three $10 debts from y'all:		−three × −ten = you gain $30

Example: A Video of People Running

People Run Forrad, Video Normal:

All normal, people running frontward: +1 × +1 = +ane

People Run Forward, merely Video in Reverse:

Looks like people running backwards: +ane × −1 = −one

People Run Backward, Video Normal:

Yous meet people running backwards: −ane × +one = −1

People Run Astern, simply Video in Reverse:

Looks similar people running forwards: −ane × −i = +1

Example: Tank Levels Ascension/Falling

The tank has 30,000 liters, and ane,000 liters are taken out every day. What was the corporeality of h2o in the tank 3 days ago?

We know the amount of water in the tank changes by −one,000 every day, and we need to subtract that iii times (to go back 3 days), so the change is:

−three × −1,000 = +3,000

The full adding is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago at that place were 33,000 liters of water in the tank.

Multiplication Table

Here is another fashion of looking at it.

First accept a play with this (explanations below):

numbers/images/mult-grid.js?min=-5&max=5

Start with the multiplication table (simply up to 4×4 volition do):

×	1	2	3	four
1	1	2	iii	4
two	ii	4	6	viii
three	3	6	9	12
4	4	8	12	16

Now run into what happens when we caput into negatives!

Let'south go backwards through zilch:

×	1	2	3	4
-4	-4	-8	-12	-sixteen
-3	-3	-six	-9	-12
-two	-2	-4	-6	-eight
-i	-i	-2	-iii	-4
0	0	0	0	0
1	1	2	iii	iv
2	2	iv	half-dozen	8
3	three	6	nine	12
4	four	8	12	xvi

Expect at the "4" cavalcade: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting iv larger each time.

Look over that table again, make sure you lot are comfortable with how it works, considering ...

... at present nosotros get further to the left, through zero:

×	-4	-3	-ii	-1	1	2	3	4
-four	xvi	12	eight	4	-4	-8	-12	-xvi
-3	12	ix	6	3	-three	-6	-9	-12
-2	8	6	iv	ii	-ii	-4	-half-dozen	-eight
-1	4	3	2	i	-1	-2	-3	-4
0	0	0	0	0	0	0	0	0
i	-iv	-three	-2	-1	ane	2	3	4
2	-8	-6	-4	-2	2	iv	half dozen	eight
iii	-12	-9	-6	-three	three	6	9	12
four	-16	-12	-viii	-4	4	8	12	xvi

We tin can follow along a row (or column) and the values change consistently:

Follow the "iv" row: it goes -sixteen, -12, -viii, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
Follow the "-4" row: information technology goes sixteen, 12, 8, four, 0, -4, -8, -12, -16. Getting 4 smaller each time.
etc...

So information technology all follows a consistent pattern!

Also see this at Real Multiplication Table.

What Near Multiplying 3 or More Numbers Together?

Multiply two at a time and follow the rules.

Instance: What is (−ii) × (−3) × (−four) ?

First multiply (−two) × (−3). Two similar signs make a positive sign, then:

(−two) × (−3) = +6

Side by side multiply +six × (−4). Two dissimilar signs make a negative sign, so:

+6 × (−4) = −24

Result: (−2) × (−iii) × (−iv) = −24

330, 1615, 1616, 1617, 3447, 3448,331, 1618, 3170, 3171