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.
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.
Multiplying with q negative does genuinely correspond to a 180° rotation around the origin in the complex plane (plus a scalar multiplication of course)
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
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.
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.
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+.