### #451

Posted 15 September 2020 - 20:00

### #452

Posted 16 September 2020 - 10:05

It's amazing that imaginary and complex numbers play such an important part in quantum mechanics - and quantum mechanics is key to explaining much of the "real world" = from your biochemistry to stars and given us vital technology such as transistors. And if anyone thinks they fully understand quantum mechanics, they don't....

One way of thinking about complex numbers is to extend the concept of a number line to two dimensions, so there are two lines at right angles which intersect at zero, like graph axes with real numbers along the x- axis and imaginary numbers along the y-axis. So the point (1,0) corresponds to the real number 1, the point (0,1) to the imaginary number i and the point (1,1) to the number 1+i. Using this picture, multiplication can be related to rotations of lines in this picture which can also help with understanding why -1 x -2 = +2.

### #453

Posted 16 September 2020 - 11:24

I find Misterioso's Mythslexia to be entirely logical in a world where you can invent i to be the square root of -1. Just like that!!

Sounds like Alice in Wonderland!!

### #454

Posted 16 September 2020 - 13:55

The negative square roots aren't something you need very often in "real world" maths.

So in a purely mathematical sense, the square roots of 4 are 2 and -2.

But in a 'real world' sense, I want to make a square raised bed with area 4m

^{2}, how long are the sides? They are 2m each. You don't need to consider the case of a raised bed with sides -2m in length. I have a friend who is a physics prof, and this is something she gets a lot with first year students - they are so keen to demonstrate the mathematical knowledge that they neglect to consider what is physically possible (and she deliberate sets such questions early on to encourage contextual thinking!)As BadStrad said, a square number is what you get when you multiply something by itself. So 1x1 = 1. -1 x -1 = 1. There is no number you can multiply by itself and get a negative answer. So i was invented to fill that gap, and has provided remarkably useful. But, not something you'll ever need to worry about if your day-to-day existance doesn't involve significant physics/engineering/maths.

Try telling anyone playing with electrical circuits that negative square roots aren't "real world"! Any periodic system that can be expressed as a sinusoid itself can be represented as the sum of two complex functions. If you're dealing with AC circuits then you're immediately plunged into the realms of complex numbers.

Then, of course, there are quaternions...!

### #455

Posted 16 September 2020 - 15:33

The negative square roots aren't something you need very often in "real world" maths.

So in a purely mathematical sense, the square roots of 4 are 2 and -2.

But in a 'real world' sense, I want to make a square raised bed with area 4m^{2}, how long are the sides? They are 2m each. You don't need to consider the case of a raised bed with sides -2m in length. I have a friend who is a physics prof, and this is something she gets a lot with first year students - they are so keen to demonstrate the mathematical knowledge that they neglect to consider what is physically possible (and she deliberate sets such questions early on to encourage contextual thinking!)

As BadStrad said, a square number is what you get when you multiply something by itself. So 1x1 = 1. -1 x -1 = 1. There is no number you can multiply by itself and get a negative answer. So i was invented to fill that gap, and has provided remarkably useful. But, not something you'll ever need to worry about if your day-to-day existance doesn't involve significant physics/engineering/maths.

Try telling anyone playing with electrical circuits that negative square roots aren't "real world"! Any periodic system that can be expressed as a sinusoid itself can be represented as the sum of two complex functions. If you're dealing with AC circuits then you're immediately plunged into the realms of complex numbers.

Then, of course, there are quaternions...!

.... that's the trouble with threads like this....

### #456

Posted 16 September 2020 - 16:13

For full disclosure, I have half of an electrical engineering degree so I had to take examinations in this stuff. That doesn't stop it from being esoteric to most people.

### #457

Posted 17 September 2020 - 16:54

But, not something you'll ever need to worry about if your day-to-day existance doesn't involve significant physics/engineering/maths.Try telling anyone playing with electrical circuits that negative square roots aren't "real world"! Any periodic system that can be expressed as a sinusoid itself can be represented as the sum of two complex functions. If you're dealing with AC circuits then you're immediately plunged into the realms of complex numbers.

Then, of course, there are quaternions...!

See the last sentence I said - that's what I meant, someone dealing with AC circuits *is* having signifcant physics/engineering/maths in the day-to-day life, and thus was exempt from me saying they didn't need to worry about i.

Like corenfa, I've taken exams in some of this stuff (for me, 75% of a physics degree ), and I did them as a leisure pursuit - this stuff intrigues and still totally baffles me, as it should! But the point of *this* thread is that it was started for those less comfortable with maths, so I think that those who may be able to open a window on that world may want to open it gently and not let the weirder corners overwhelm anyone until they are ready and willing to be overwhelmed. Otherwise, it's a bit like bringing out some complicated four-part Bach for analysis at someone's second music theory lesson