The rules here require you to show evidence of your own attempt/work before we help.ChiralSuperfields said:
Here the dot "." means multiplication. So 0.1+1.0 stands for 0x1+1x0.ChiralSuperfields said:
This is indeed a bit strange. However, you must be careful with complex numbers since some rules you are used to by real numbers do not apply anymore. Please readChiralSuperfields said:
The product after ii could as well have been written as (0 + 1i)(0 + 1i). All the author is doing here is carrying out the four multiplications: ##0 \cdot 0, 1 \cdot 1 \cdot i^2, 0 \cdot 1 \cdot i, 1 \cdot 0 \cdot i##. Here ##i^2 = -1##, so this seems like a lot of work to show that i times itself equals -1. It's also a circular argument, since to show that ##i \cdot i## (or ##i^2##) equals -1, the argument above uses the fact that ##i^2 = -1##.
In some countries using period for multiplication is the norm. It is also not circular if complex numbers are defined as pairs of real numbers written as ##a+bi## and operations defined by:Mark44 said:
The circularity I referred to was in the expansion of ##i \cdot i## where one of the partial products was ##(1 \cdot i)(1 \cdot i) = 1 \cdot i^2##. IOW, to find ##i \cdot i## one needs to know that ##i \cdot i = -1##.martinbn said:
No, that was my point. The definition of multiplications is ##(a+bi)\times(c+di)=(ac-bd)+(ad+bc)i##. Also ##i## is short for ##0+1i##. ThusMark44 said:
I understand your point. My concern is that in evaluating ##(a+bi) \times (c+di)##, one of the terms in the addition is -bd, which comes from ##b \cdot d \cdot i^2##, where ##i^2## is implicitly replaced by -1. This means that this equation is implicitly using the definition of ##i^2## being equal to -1.martinbn said:
Worse than confusing, it is not true. Certainly, ##ii = (0+i)(0+i) = -1##, but the orange part is ##-1 + 1.1i##, which does not fit in with the rest of it. I can't think of any interpretation or typo that might fit in.ChiralSuperfields said:
That confused me too, but Mark figured it out:FactChecker said:
Mark44 said:
I think that @martinbn 's posts 3 and 6 explain the interpretation and the motivation of the OP.berkeman said:
No, there is no evaluation. It is a defining identity. The complex numbers are defined as the set of expressions ##a+bi## with ##a,b## real numbers and operations 'addition' and 'multiplication' given by the formulas I wrote.Mark44 said:
Yes. that is the way that the OP would make sense. There are geometric and algebraic motivations for the complex plane and number system that would give the multiplication formula ##(a+bi)(c+di)=(ac-bd)+(ad+bc)i##martinbn said:
The advantage of that algebraic perspective is the different mindset. One does not associate ##x=\sqrt{-1}## like one would when using ##\mathrm{i}.## Nobody would write a line like that:WWGD said:
Yes, kind of a nightmare dealing with the branching alone.fresh_42 said:
I learned the branching thing with a comparison to that:WWGD said:
No German porn, please.fresh_42 said:
And for good reason, since the properties of radicals explicitly disallow equating ##\sqrt{a\cdot b}## with ##\sqrt a \cdot \sqrt b## when both a and b are negative reals.fresh_42 said:
If it is a teachers lecture notes, have him/her firedhutchphd said:
hutchphd said:
You are both assuming that this was the students' first exposure to complex numbers. May be they already knew complex numbers and this is a way of introducing them to some more abstract algebra through something they know. Here is a set and operations, now we deduce some properties.malawi_glenn said:
Considering OPs previous posts, this is a reasonable assumption tbh.martinbn said:
... which is why I first recommended readingMark44 said:
Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the property that i² = -1.
The imaginary unit i is defined such that its square is -1. This definition extends the real number system to the complex number system, allowing for solutions to equations that have no real solutions, such as x² + 1 = 0.
To simplify expressions involving complex numbers, you combine like terms (real parts with real parts, imaginary parts with imaginary parts) and use the property that i² = -1 to reduce higher powers of i. For example, i³ = i² * i = -1 * i = -i.
Without seeing the specific expression in orange, it's likely that it involves the property of the imaginary unit i. If the expression includes i², it simplifies to -1 because by definition, i² = -1.
Complex numbers are used in various fields such as engineering, physics, and applied mathematics. They are essential in the study of electrical circuits, signal processing, fluid dynamics, quantum mechanics, and many other areas where waveforms and oscillations are analyzed.