undefined

points

[-]

i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.

"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.

https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.

https://en.wikipedia.org/wiki/Dimensionless_quantity

by ttoinou5 hours ago|

parent|

[-]

Yes I agree, I was just arguing against "i=0+1*i so it's by definition a complex number" which is a self referential definition

by chongli7 hours ago|

prev|

[-]

If you take this tack, then 0 and 1 are not numbers either.

by ttoinou5 hours ago|

parent|

[-]

i is not a real number, is not an integer, is not a rational etc.

You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number

0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers

by chongli4 hours ago|

parent|

[-]

The simple fact you wanted to write this:

i=0+1*i

Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.

Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.

Comparing i with a unary operator like - shows the difference:

i*i=-1 makes perfect sense

-*-=???? does not make sense

by ttoinou3 hours ago|

parent|

[-]

i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1

  i*i=-1 makes perfect sense

This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles

by chongli2 hours ago|

parent|

[-]

There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].

[1] https://en.wikipedia.org/wiki/Peano_axioms