How To Multiply Negative Number

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Multiplying Negatives

When We Multiply:

		Example
×	two positives make a positive:	iii × 2 = 6

×	two negatives make a positive:	(−3) × (−2) = 6

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

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

Yep indeed, two negatives brand a positive, and we volition explicate why, with examples!

Signs

Let's talk about signs.

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

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

Example: five is actually +5

And we can put () effectually the numbers to avoid defoliation.

Example: 3 × −2 tin be written as (+3) × (−2)

2 Signs: The Rules

multiply like signs

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

Case: (−2) × (+5)

The signs are − and + (a negative sign and a positive sign), and then they are unlike signs (they are unlike to each other)

So the effect must exist negative:

(−2) × (+5) = −10

Case: (−iv) × (−3)

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

So the result must be positive:

(−four) × (−three) = +12

Why does multiplying two negative numbers brand a positive?

Well, first at that place is the "common sense" explanation:

dog does not want brocolli

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

Just when I say "Exercise not eat!" I am maxim the opposite (negative).

At present if I say "Practise Not not eat!", I am saying I don't want you lot to starve, so I am back to saying "Eat!" (positive).

So, ii negatives make a positive, and if that satisfies you, and then yous don't need to read any more than.

Direction

It is all nigh management. Remember the Number Line?

Well hither we have Infant Steven taking his first steps. He takes two paces at a fourth dimension, and does this iii times, so he moves 2 steps ten iii = 6 steps forrard:

multiply 2 x 3

Now, Baby Steven tin can too step backwards (he is a clever picayune guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this 3 times:

multiply -2 x 3

Once again Steven'due south Dad puts him dorsum at the start, but facing the other style. Steven takes two steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

multiply 2 x -3

Back at the start once more (thanks Dad!), nonetheless facing in the negative direction, he tries his backwards walking, in one case once again taking two steps at a time, and he does this 3 times:

multiply -2 x -3

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

Endeavour it yourself! Try walking forwards and backwards, and so again but facing the other direction.

Play With It

But perhaps you lot would like to see information technology in action? Use the sliders beneath:

numbers/images/number-line-mult.js

More Examples

Example: Money

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

Example: A Video of People Running

People Run Forward, Video Normal:

All normal, people running frontwards: +1 × +one = +1

People Run Forward, only Video in Contrary:

Looks like people running backwards: +1 × −1 = −i

People Run Backward, Video Normal:

You come across people running backwards: −ane × +one = −ane

People Run Backward, merely Video in Reverse:

Looks like people running forwards: −1 × −1 = +one

Instance: Tank Levels Rising/Falling

The tank has 30,000 liters, and 1,000 liters are taken out every 24-hour interval. What was the corporeality of water in the tank 3 days ago?

We know the amount of h2o in the tank changes by −1,000 every mean solar day, and we demand to subtract that 3 times (to go back 3 days), so the modify is:

−3 × −i,000 = +3,000

The full adding is:

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

So 3 days agone there were 33,000 liters of h2o in the tank.

Multiplication Table

Hither is another mode of looking at it.

Beginning accept a play with this (explanations below):

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

Start with the multiplication table (but upward to iv×4 will do):

×	i	ii	3	4
1	i	2	3	4
ii	2	4	half dozen	8
3	3	6	9	12
four	4	8	12	16

Now see what happens when nosotros head into negatives!

Let's become backwards through zero:

×	1	2	3	iv
-4	-4	-eight	-12	-sixteen
-three	-three	-vi	-nine	-12
-2	-2	-iv	-6	-8
-1	-1	-2	-iii	-four
0	0	0	0	0
1	one	2	iii	iv
2	2	4	6	8
3	three	6	ix	12
iv	4	viii	12	16

Look at the "four" column: it goes -16, -12, -8, -four, 0, 4, 8, 12, 16. Getting four larger each time.

Look over that table again, brand certain you are comfortable with how information technology works, because ...

... now we go further to the left, through nothing:

×	-four	-iii	-two	-1	1	two	3	4
-4	16	12	8	4	-4	-8	-12	-xvi
-3	12	9	6	3	-three	-6	-nine	-12
-2	8	6	4	2	-ii	-4	-6	-8
-1	4	3	2	1	-1	-2	-3	-iv
0	0	0	0	0	0	0	0	0
one	-4	-3	-2	-ane	1	2	iii	4
2	-8	-vi	-four	-ii	2	four	half dozen	8
iii	-12	-9	-6	-three	iii	6	9	12
4	-16	-12	-eight	-4	4	8	12	16

Nosotros tin can follow along a row (or column) and the values modify consistently:

Follow the "four" row: it goes -16, -12, -8, -four, 0, four, eight, 12, 16. Getting 4 larger each time.
Follow the "-4" row: it goes xvi, 12, 8, 4, 0, -four, -viii, -12, -sixteen. Getting four smaller each fourth dimension.
etc...

Then information technology all follows a consequent pattern!

Too run into this at Real Multiplication Table.

What Nearly Multiplying 3 or More Numbers Together?

Multiply two at a fourth dimension and follow the rules.

Example: What is (−ii) × (−3) × (−4) ?

Starting time multiply (−2) × (−three). 2 like signs make a positive sign, so:

(−two) × (−3) = +6

Adjacent multiply +six × (−4). Two unlike signs make a negative sign, and so:

+six × (−iv) = −24

Issue: (−two) × (−iii) × (−4) = −24

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