Why do quaternions not only use two imaginary numbers. Can we not simplify quaternions $$q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \tag{1}$$ to the form $$ \begin{align} q &amp; = a + b\mathbf{i} + (c + d\mathbf{i})\mathbf{j} \tag{2}\\   &amp; = e + f\mathbf{j} \end{align} $$ Where $e,f \in \mathbb{C}$. Thus we only introduce one new imaginary number. Where we note that $\bf{ij} = k$, which holds true. This would make quaternions seem more like complex numbers and we only need to define the interactions between $\bf i$ and $\bf j$. The same would also hold true for octonions being represented as $s = g + h\mathbf{k'}$ where $g, h \in \mathbb{H}$. Where we see that we now have only 3 imaginary numbers instead of 7. This 'factorized' representation seems to simplify things and make these numbers look more familiar.
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Why is  quaternion algebra 4D and not 3D?  Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.    $ i^2 = j^2 = k^2 = ijk = -1 $  Using $1, i, j,$ and $k$ as the base (where complex uses $1$ and $i$ (or $j$ if you are an EE)) which results in a 4-axis space.  Why is there no 3D algebra?
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If I can prove that $ij = k$, given that $i^2 = j^2 = k^2 = -1$, then it will be easy to prove the other quaternion formulas. However, I'm having a lot of trouble getting past this step. I started by setting up the following equation, such that $(a, b, c, d) \in \mathbb{R}$  $ij = a + bi + cj + dk$  Obviously, I need to prove that $a = b = c = 0 \land d= 1$. I've done all sorts of wacky things with this equation and I haven't gotten any closer to proving that $ij = k$. I would greatly appreciate it if somebody could point me in the right direction. Thanks in advance!
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$z_1$, $z_2$, and $z_3$ are 3 complex numbers. Prove that if they represent the vertices of an equilateral triangle then $z_1 + \omega z_2 + \omega^2 z_3 = 0$ where $\omega$ is a 3rd root of unity. Any help would be thoroughly appreciated.
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Hi I'm having trouble understanding why the quaternionic group cannot be written as a semi-direct product in a non-trivial way. Is there a proof of why this is the case?
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I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real explanations of the benefits of using them.  I'm wondering what exactly can be done with Quaternions that can't be done as easily (or easier) using more tradition approaches, such as with Vectors?
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What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples and explanations would be appreciated.
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So the question is in the title. It's easy to prove when we know that there are real numbers in it and the dot product is standard. But why this works in the general case - when there are complex numbers inside and the dot product is something else?
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Two complex numbers $z_{1}$ and $z_{2}$ are taken such that    $|z_{1} + z_{2}| = |z_{1} - z_{2}|$ and $z_{2}$ is not $0$.   Show that $z_{1}/z_{2}$ is purely imaginary, i.e. it has no real part.   So they are both absolute values so $|z_{1} + z_{2}| = |z_{1} + z_{2}|$ is also true so how can you ever solve it if they are equal or is my assumption false? also i have tried to turn in into $a + bi$ and got to $a_{1}\cdot a_{2} = -b_{1}\cdot b_{2}$, but i dont see how this will help me to get to prove the statement. Also what it means that $z_{2}$ is not $0$ confuses me because $b$ or $a$ can still be $0$ then but just not both at the same time.  Can someone give me a hint without giving the whole answer? So i can still figure the rest out myself. Just a hint to be able to continue.
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Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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Is there a method to find which numbers to use when simplifying quadratics?   For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: $x^2 + 7x – 6$, then it gets very hard to think of a valid simplification.  EDIT: Here's an example: $(x^2 + 5x + 6)/(x + 2)$. When simplifying the first half we get $(x + 2)(x + 3)/(x + 2)$, then we can cancel out $(x + 2)$ and the answer would be $(x + 3)$. The question is: How to always find the right numbers which to put in? (in this example they where 2 and 3)
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At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?  Why is the concept even useful?
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Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from.
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Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?  This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.
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Suppose $a$ and $b$ are complex numbers and both transcendental over $\mathbb Q$. I am wondering why $ab$ and $a+b$ can not both be algebraic.  Thanks for any help.
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I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?
18930
The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?   I can identify two different settings where the answer may be different:   It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical electrodynamics. Can we say the same thing about quantum mechanics either way for sure?   Could this be a difference in quantum mechanics over the classical picture?
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Should I write complex numbers as $a+ib$ or $a+bi$? Maybe there are certain cases where I should write it one way and other cases where I should write it the other way. What do you think?