That’s surprisingly accurate, as people here are highlighting (it makes geometrical sense when dealing with complex numbers).
My nephew once asked me this question. The way that I explained it was like this:
- the friend of my friend is my friend; (+1)*(+1) = (+1)
- the enemy of my friend is my enemy; (+1)*(-1) = (-1)
- the friend of my enemy is my enemy; (-1)*(+1) = (-1)
- the enemy of my enemy is my friend; (-1)*(-1) = (+1)
It’s a different analogy but it makes intuitive sense, even for kids. And it works nice as mnemonic too.
I teach maths and one of the analogies is use is watching a film of someone walking forwards and backwards. If you play the film forwards (multiplying by positive), you can see the person walking forwards and backwards as normal. If you play the film backwards (multiplying by negative) you see the opposite. So multiplying by negative reverses whatever was happening before. Hard to put into words but the visuals (hopefully) seem to explain it well enough.
That’s exactly what eventually helped me understand it.
To multiply the negative by a negative is like an instruction to “reverse the circumstance that created the negative and then keep ‘reversing’ forward”, so to speak.
You come across a hole in the ground. You see a shovel and 5 piles of dirt. That hole in the ground represents where that dirt used to be.
You can “add” more depth to the hole by digging, i.e. continuing to remove dirt and create more piles.
But you can also reverse what was done by “un-digging”, I.e. putting the dirt back into the hole.
So if you “un-dug” the hole with the 5 piles of dirt, you’d have 0 piles, and 0 holes.
But if you “un-dug” the hole 5 times in a row, you’ve filled the hole and started creating a pile on top of it with dirt from somewhere else.
It’s a different analogy but it makes intuitive sense, even for kids.
Its good maths but terrible realpolitick.
Realpolitik is terrible no matter what.
My math teacher in middle school explained it with love/hate, but same set up.
If you hate to love you’re a hater If you love to hate you’re a hater
Oh this is a really cool way to think about it! Thanks for sharing
This is basically the staple way of explaining the topic in my country. It was a very bizzare concept for 13 year old me so it made understanding it a lot easier.
Sorry for the question, but where are you from? I learned this with my mother, so I don’t know if it’s something common here (Brazil) or something that she picked from her Polish or Italian relatives.
I mean, you can negotiate a peace agreement and free trade deals, you don’t have to be enemies.
Lmao not gonna lie, this would be a very intuitive way of teaching a kid negative values.
How is multiplying akin to rotating?
The 180 deg rotation indicates multiplying by negatives. It’s a good analogy to represent change to the opposite side. Which multiplying with negatives does, the number goes from one side of 0 on the number line to the other side.
It was used in my. complex numbers class - multiply by i means rotate 90 degrees on complex plane
and multiplying by i twice means you rotate 90 degrees twice. So 180 degrees. Multiplying by i twice means multiplying by i^2 = -1.
So multiplying by -1 is a rotation by 180 degrees
I think the problem with this example is, while it provides a simple/visual thing to help people wrap their head around the idea of it, it still doesn’t really explain it.
When you turn around, you are spinning around a central point, in a circular motion, and when you have a circle, you will always end up back at the start. You start at 0°, and if you keep adding degrees, eventually you will hit 365°, and one degree more puts you back to 0° off from where you started. You do not have to reverse direction to get back to 0°
But with negative numbers you’re trying to explain a line, and that line goes in both directions infinitely. There is no point on the positive side of the line where you get teleported back to 0. You start at 0, keep adding numbers, and it’s going to keep going down that line forever until you start subtracting.
The problem is usually more about the person not understanding what a negative number actually represents. They just think of them as regular numbers that happen to exist on the other side of 0 on the number line. So multiplying by a negative number gets treated the same as multiplying by a positive, just with a little dash next to the result.
Real numbers are 1-dimensional in that they all fit in just one continuous line, so there are only 2 directions you can face in that line - forwards and backwards - hence only 2 possible values for rotation (quite literally it’s a binary option).
So when mapping the rotation around on a plane (i.e. a 2D rotation) to a 1D rotation, because the 2D rotation is a range of possibilities you need to pick 2 and only 2 positions out of the infinite possibilities and they both must obbey they rule that you rotate in 2D from one to the other one you end up facing the opposite direction.
As it so happens any 2 numbers for a 2D angle of rotation that differ by 180 degrees or PI radians obbey both rules so any such pair of numbers can be used. Because in 2D rotations, there is the property that any rotation angle is equivalent to any angle which differs from it by a multiple of +/- 360° that gives you more rotation value pairs which have different numbers but represent the same rotation.
For simplicity, the tendency is to use 0° and 180°, but anything that obbeys the rules above would work, say -270 and +90, or 96 and 276 as long as you can form a straight line in the 2D plane for that rotation passing both angles and the center of rotation you can use them to express 1D rotations.
(-1)*(-1) = eiπ*eiπ = ei2π = 1
We’re looking for ELI5, not ELImathematician
Lol, yeah good luck explaining ‘e’ to a child who doesn’t understand basic multiplication.
Idk how they do it these days, but when I was a kid we learned the alphabet several years before multiplication. /j
Multiplying with q negative does genuinely correspond to a 180° rotation around the origin in the complex plane (plus a scalar multiplication of course)
Very intuitive to kids
I take your point, but honestly I’d bet many would be ready to learn about complex numbers a lot earlier if they were taught in this way.
Having such a memorable physical analogy “because I said so” is already miles better than the purely abstract “multiplying negatives makes a positive because I said so”, even if it still doesn’t mean you could teach extremely high level maths to six year olds.
Agreed. I’m trying to keep the reigns on an 11 year old, and we frequently talk both in what I would say is abstract. Also have to keep it somewhat grounded, because skipping multiple grades in math does not mean you will understand some things. Absolute value was an interesting conversation, and to be fair so was multiplying negatives.
I was already trying to visualize multiplying as a circle in my head and something clicks but cant grasp it.
Now reading that apparently There is a real mathematical link i am dying to learn more. Do you know of an online visualizer/simulation that helps showing what you just said?
Honestly, the best online resource I know of are the 3blue1brown videos on complex numbers
Any tool risks confusing you more, since multiplying in the complex plane can act quite unexpectedly when you move outside the real line for both parameters
Had a look around and this will quickly become one of my favorite media channels! Thanks!
https://www.3blue1brown.com/lessons/eulers-formula-via-group-theory
Minute 12 in the video is the most relevant, but the whole lesson is worth going through.
3blue 1brown is a great channel for challenging how you think about math in a beautifully animated fashion.
I just had a look on their channel. I think my old classmates would cringe if they knew how excited i got seeing these thumbnails and titles.
All my initial scientific inspiration have gotten sucked dry in the meat mill that is the education system, but living in the age of educational internet videos is big healer.
I got vertasium and steve mould. Kurtzegesagt is mandatory for everyone by now i hope, i still follow Vsauce but i miss Michael. Got any other recommendations?
Mathloger, numberphile, computerphile, Sixty symbols: more good math/computer science theory channels
applied science, breaking taps: truly amazing “garage” engineering. They take on projects that you would normally expect to take a specialized lab.
alpha phoenix: his expertise is in materials science but he does delve a bit into electromagnetic questions
Mr P Solver: solving interesting problems computationally in pthyon
Eevblog: good electrical engineering insights with a nice Australian accent
Practical engineering: all the civil engineering questions you never knew you had
Stuff made here: what happens if a robotics expert has a generous fun projects budget and never sleeps
Tropical tidbits: discussion of the meteorology of tropical storms and hurricanes as they happen with none of the weather reporting sensationalism
I’m sure I’m missing some, but that should be a big enough list to add many hours to your watch list.
I have a physics degree, and 3 blue 1 brown’s latest videos on light are amazingly presented in comparison to the vast majority of lectures I’ve sat through. It makes me hopeful that online video sharing can help improve pedagogy and not just be clickbait nonsense.
Maybe this help https://www.nagwa.com/en/explainers/280109891548/
Or you can search Argand Diagram
Fun fact: exponents and multiplication DO work like rotation … in the complex domain (numbers with their imaginary component). It’s not a pure rotation unless it’s scalar, but it’s neat.
I know I explained that the worst ever, but 3blue1brown on YT talks about it and many other advanced math concepts in a lovely intuitive way.
Because it works
It’s the sign rather than the absolute value of the multiplicator that defines a rotation of 180° for
-or 360°/0° for+.
“It absolutely, definitely makes sense”
Turn Around. Every now and then I get a little bit lonely and you’re never coming 'round…
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A pretty general explanation is that a number consists of an length and an angle on the number line. Positive numbers have angle = 0. Negative numbers have angle = pi (or 180° if you want to work with degrees instead of radians).
Multiplication is an operation where you add together the angles to retrieve the resulting angle and multiply together lengths to get the resulting length (yes, kinda recursive, but we’re only working with purely positive numbers here).
So 3 * (-3) means
Length = 3 * 3 = 9
Angle = 0 + pi = pi (or 0 + 180° = 180°)Of course this is very pedantic, but it works in more complex scenarios as well (pun intended).
Imaginary numbers have angle pi/2 (or 90°) or 3pi/2 (or 270°). So if you for instance want to find the square root of i, you can solve it by finding the length:
1 = x * x
And angle:
pi/2 = y + y
(can use modulus 2pi to acquire 2 solutions here)Solving the equations and resolving the real and imaginary part with trigonometry, we get
1/sqrt(2) + 1/sqrt(2)*i
And
-1/sqrt(2) - 1/sqrt(2)*i
It’s all just circles all the way down
I’m sure circles fit into string theory somehow too.
I’ve never thought of numbers having a direction in a number line, that’s great. Thank you for explaining it this way!
If you are curious about the math logic side of this like I was, here’s the explanation.
Multiplying is just like addition.
3 * 3 = 3 + 3 + 3 = 9
Simple enough but what if one is negative?
3 * (-3) = (-3) + (-3) + (-3) = -9
Also easy, all we changed was the signs of the 3’s being added together. But what do you change if you make both of them negative? The only thing left to change is the operation sign. Thus multiplying two negatives is like subtracting negatives.
(-3) * (-3) = - (-3) - (-3) - (-3) = 3 + 3 + 3 = 9
Notice that I placed a negative sign in front of the first (-3). That first one has to be subtracted as well so you can imagine a zero in front of the operation.
Edited: Some formatting.
Copy pasted from my other comment:
This doesn’t work if you have to deal with multiplication of numbers that are not integers. You can adjust your idea to work with rational numbers (i.e. ratios of integers) but you will have trouble once you start wanting to multiply irrational numbers like e and pi where you cannot treat multiplication easily as repeated addition.
The actual answer here is that the set of real numbers form a structure called an ordered field and that the nice properties we are familiar with from algebra (for ex that a product of two negatives is positive) can be proved from properties of ordered fields.
Before i ask my question, know that my math is all the way in the back of my head and i didnt get too far in math at school.
Wdym irrational numbers dont work? -3 * -pi would be the same as 3*pi, no?
I always assumed if all factors of the multiplication are negative, it results in the same as the positive variant, no matter the numbers ( real, fractal, irrational, … )3 pi = pi + pi +pi
Sure thats okay, but what about e * pi?
Multiplying two negative irrational numbers together will still give you a positive number, it’s just that you can’t prove this by treating multiplication as repeated addition like you can multiplication involving integers (note that 3 is an integer, 3 is not irrational, the issue is when you have two irrationals).
So, for example with e * pi, pi isn’t an integer. No matter how many times we add e to itself we’ll never get e * pi.
Try it yourself: Assume that we can add e to itself k (a nonnegative integer) times to get the value e * pi. Then e * pi = ke follows by basic properties of algebra. If we divide both sides of this equation by e we find that pi=k. But we know k is an integer, and pi is not an integer. So, we have reached a contradiction and this means our original assumption must be false. e * pi can’t be equal to e added to itself k times (no matter which nonnegative integer k that we pick).
Thanks for that! Helped understand it a bit more, i think. So its a case of it not working on irrational numbers, its just that we cant prove it because we cant calculate the multiplication of 2, right? Somehow, my mind has issues with the e*pi = ke. Id say that ke = e * pi is impossible because k is an integer and pi isnt, no? It could never be equals, i think.
So its a case of it not working on irrational numbers, its just that we cant prove it because we cant calculate the multiplication of 2, right?
The issue is the proving part. We can’t use repeated addition trickery (at least not in an obvious way) to show a product of two irrational negative numbers is positive. It’s definitely still true that a product of two negative numbers is positive, just that proving it in general requires a different approach.
Somehow, my mind has issues with the e*pi = ke. Id say that ke = e * pi is impossible because k is an integer and pi isnt, no? It could never be equals, i think.
Yes this is correct. The ke example is for a proof by contradiction. We are assuming something is true in order to show it forces us to be able to conclude something ridiculous/false. Since the rest of our reasoning was correct, then it must have been our starting assumption that was wrong. So, we have to conclude our starting assumption was wrong/false.
I think all they mean is you can’t write it out since irrational numbers have no end.
You’re correct in that the principle still applies in exactly the same way.
Fun fact… a formal definition of irrational numbers didn’t exist until the 1880s (150+ years after Newton died). There were lots of theories before that time (including that they didn’t exist) and they were mostly ignored. Iirc, it was Euler’s formal definition of complex numbers and e (an irrational number) that led to renewed interest.
There are some subtleties to this particular topic that are worth mentioning. I would be careful to distinguish between constructing vs defining here.
The usual definition of the irrationals works roughly like this:
You have a set of numbers R which you call the real numbers. You have a subset of the real numbers Q which you call the rational numbers. You define a real number to be irrational if it is not a rational number.
This is perfectly rigorous, but it relies on knowing what you mean by R and Q.
Both R and Q can be defined “without” (a full) construction by letting R be any complete ordered field. Such a field has a multiplicative identity 1 by definition. So, take 0 along with all sums of the form 1, 1+1, 1+1+1 and so on. We can call this set N. We can take Z to be the set of all elements of N and all additive inverses of elements of N. Finally take Q to be the set containing all elements of Z and all multiplicative inverses of (nonzero) elements of Z. Now we have our R and Q. Also, each step of the above follows from our field axioms. Defining irrationals is straightforward from this.
So, the definition bit here is not a problem. The bigger issue is that this definition doesn’t tell us that a complete ordered field exists. We can define things that don’t exist, like purple flying pigs and so on.
What the dedekind cut construction shows is that using only the axioms of zfc we can construct at least one complete ordered field.
The snark in that first reply is glorious.
TIL math logic
I wrote about this after pondering in my blog: http://hypotheticallyy.blogspot.com/2023/08/why-is-negative-times-negative-positive.html
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It’s not that hard.
If you have -3 -3s and I give you one, you now only have -2 -3s. If you want to get to a total of -6, I have to hand over 4 more -3s to get there, the first 3 of them just being what’s needed to get you to 0 and out of deficit. Now you get to hold onto the next two I hand over, and now you have 2 -3s which total -6. But that’s 15 worth of -3s I had to hand over to get you there and -6 + 15 = 9, like -3 × -3 does too.
Negative numbers aren “real”. Like 0, they’re just a concept used to represent something, deficit.
umm
Headhurts
I like the greentext explanation better.
This doesn’t work if you have to deal with multiplication of numbers that are not integers. You can adjust your idea to work with rational numbers (i.e. ratios of integers) but you will have trouble once you start wanting to multiply irrational numbers like e and pi where you cannot treat multiplication easily as repeated addition.
The actual answer here is that the set of real numbers form a structure called an ordered field and that the nice properties we are familiar with from algebra (for ex that a product of two negatives is positive) can be proved from properties of ordered fields.
Don’t confuse the wording “set of real numbers” here, this is just the technical name for the collection of numbers people use from elementary algebra on through to calculus.
