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 consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (√-1).
To add or subtract complex numbers, you simply combine the real parts and imaginary parts separately. For example, (2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i.
The expression in orange is most likely referring to the imaginary unit, i. When squared, i becomes -1. This is because i is defined as √-1, and when squared, it becomes -1.
Complex numbers are used in many fields of science and engineering, such as in electrical engineering, quantum mechanics, and signal processing. They are also used in everyday applications, such as in the design of computer graphics and in financial modeling.
Complex numbers allow us to represent and manipulate quantities that cannot be expressed using real numbers alone. They are especially useful in solving equations and problems involving waves and oscillations, as well as in analyzing and understanding complex systems.