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Imagine a highschool freshman walks up to you and asks you what hypercomplex numbers are. Explain to her, in a fair amount of detail, the different types of hypercomplex numbers in a way that any person can understand.

This was something asked to me by one of my friends. I don't know what Hypercomplex numbers are well enough to explain it. Neither does my friend and that's the reason he asked my that question. Wikipedia and other resources have been rather inefficient in delivering even the basic gist of it to us.

We are merely curious to what this extension is basically. Can you please explain it to us?
A simple glance of what it is might suffice.

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    $\begingroup$ It is done in the same way as how to convince a newborn baby of the correctness of Andrew Wiles' proof of Fermat's last theorem. $\endgroup$ – azimut Nov 11 '13 at 12:38
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    $\begingroup$ I'd rather tell her to go and study seriously geometry, basic calculus and trigonometry and, if she wants, after doing that I might be mulling to use some of my precious time to come up with as good a plan to explain her hypercomplex numbers as I can design. $\endgroup$ – DonAntonio Nov 11 '13 at 12:55
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    $\begingroup$ Why would children be at all interested in hypercomplex numbers? $\endgroup$ – lhf Nov 12 '13 at 13:32
  • $\begingroup$ @lhf: I was just as surprised as you were. $\endgroup$ – Nick Nov 12 '13 at 14:54
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    $\begingroup$ How would you explain complex numbers to hyperchildren? $\endgroup$ – Asaf Karagila Nov 16 '13 at 16:04
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Kids of a young age can understand very easily hypercomplex numbers. Several weeks ago, I demand to my 13 year old boy to find the square root of -1. After some thought, he says that it was impossible. He knew the concept of the real line; I explained then that we can put a line perpendicular to that one and call the numbers on that line the ‘i’ numbers. Then I introduced the idea that multiplying by ‘i’ is a rotation of 90 degrees. When I demanded him to multiply 1 by ‘i’ two times, his eyes staring at me with an admirable joyfulness: he had understood that i^2 was -1.

The last weekend I explained him, and his 11 year old brother, that we can build three different planes in the same fashion. The first, with an ‘i’ letter, is called the complex plane and points in it are ‘complex numbers’. The second, with a perpendicular line labeled with an ‘e’ such that e^2=0 is called dual plane and the points in it are called ‘dual numbers’. Finally we can build a third plane with a Greek letter ‘iota’ on it, such that (iota)^2=1 and points in the plane are called ‘split numbers’. Then, I explained that each point is represented by two numbers, but the rules for multiplying the pairs of numbers are found by the geometry in these planes.

After that, when I mentioned that we could build pairs of pairs, and find rules for adding and multiplying these ‘pairs of pairs’. They were amazed but very interested. All the afternoon we were counting the classes of possible combinations. My 13-yo found that there are 19,683 classes of numbers with 512 real numbers (in pairs). We call all these classes the Quintocento-dodekaions.

p.s.: Two weeks after his first encounter with complex numbers, my son demanded me for the square root of 'i' and with very little effort he found the correct operation to find this number.

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There is some history behind this.

In the beginning there was ${\mathbb N}=\{1,2,3,\ldots\}$, with $+$, $\cdot$, and $<\>$.

Certain simple equations of the form $a+x=b$ couldn't be solved in ${\mathbb N}$, so they invented ${\mathbb Z}$ containing also $0$ and the negative numbers.

Certain simple equations of the form $ax+b=c$ couldn't be solved in ${\mathbb Z}$, so they invented ${\mathbb Q}$.

Certain simple equations like $x^2=2$ couldn't be solved in ${\mathbb Q}$, nor was there a representant for the area of a unit disk. So they invented ${\mathbb R}$.

Certain simple equations like $x^2+1=0$ couldn't be solved in ${\mathbb R}$, so they invented ${\mathbb C}$, the system of complex numbers. Each complex number can be written in the form $x\>1+y\>i$ with real $x$, $y$ and a special complex number called $i$.

Hamilton then tried in vein to set up a "hypercomplex" number system where each "number" would be of the form $x\>\vec i+y\>\vec j+z\>\vec k$, where $\vec i$, $\vec j$, $\vec k$ are the basis vectors used in elementary vector algebra of ${\mathbb R}^3$. He didn't succeed, but he realized that such a system is possible when the individual hypercomplex numbers are of the form $t\>1+x\>\vec i+y\>\vec j+z\>\vec k$ with $t$, $x$, $y$, $z$ real, and if the operations $+$ and $\cdot$ are appropriately defined. In this way the first true hypercomplex number system, called the quaternions, was born. Apart from the commutativity of multiplication all "rules of algebra" are valid in this system.

It is then natural to ask, for which dimensions $n$ apart from $1$, $2$, $4$ such a system $S$ with "numbers" $\sum_{k=1}^n x_k e_k$, where $\>x_k\in{\mathbb R}$ and the $e_k$ are certain special numbers of $S$, can be set up such that one has an addition and a "reasonable" multiplication in $S$. It is one of the deep theorems of $20^{\rm th}$ century mathematics that there is just one more such system, the Cayley octonions with $n=8$; but associativity of multiplication is no longer present in this system.

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Quaternions were created by Hamilton in the nineteenth century motivated by geometric reasons. The first thing you need to know before trying to show someone else the role of quaternions in mathematics is that complex numbers weren't created because someone needed to solve $x^2+1=0$.

No one had ever cared for it, since roots meant touching the $x$-axis. The parabola formed by $y=x^2+1$ do not touch the $x$-axis and there are no roots, great, this is exactly what we see. The $\mathbb{C}$ story only starts with Bombelli's thought on Cardano's formula for cubics. You can see details here, the first chapter of Visual Complex Analysis, by Needham.

After some centuries lurking around, complex numbers gained a geometric meaning: multiplying a number $z$ by $i$ rotates it $90^\circ$ counterclockwise, $zw$ is seen as a sequence of dilatations and rotations, everything is concisely explained by $z=a+bi=|z|\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right)=|z|e^{i\theta}$. Of course, these discoveries were historically highly non-trivial and are the hard-won result of a great struggling of people like Bombelli, Euler, Argand and Gauss while trying to understand these weird numbers. At this point, the development of complex analysis, proofs of the fundamental theorem of algebra and other discoveries made complex numbers not so unnatural after all.

What Hamilton was trying to find was an extension of complex numbers by adding a new imaginary unit $j$, such that $i^2=j^2=-1$, but $i\neq j$. There "trinomials" should describe the $\text{3D}$ space by the same means complex numbers describe the plane: by dilatations and rotations. He tried for many years to build a consistent algebra using the three coordinates of space, but he couldn't. The fundamental reason is that you need more than three numbers to represent these operations in space. As Needham puts it:

However, a fundamental problem does arise when we try to represent these dilative rotations as points (or vectors) in space. By analogy with complex multiplication, we wish to interpret the equation $Q_1\circ Q_2=Q_3$ as saying that the dilative rotation $Q_1$ maps the point $Q_2$ to point $Q_3$. But this interpretation is impossible! The specification of a point in space requires three numbers, but the specification of a dilative rotation requires four: one for the expansion, one for the angle of rotation and two for the direction of the axis of the rotation.

When passing from the real numbers to complex numbers, you lose things as order and a careless using of roots and logarithms, but you gain an awesome lot of amazing properties: the fundamental theorem of algebra, complex analysis, a better understanding or power series and so on. In general, when widening an algebraic structure, this trade-off happens. Getting to quaternions is no different: in order to describe the space, you should give up commutativity of quaternions' multiplication, since rotations are non-commutative in $\text{3D}$.

Quaternions are, after all, members of the $\text{4D}$ space of dilative rotations in the $\text{3D}$ space. Hamilton made tremendous effort in promoting these numbers, but vector algebra is often cleaner. If I am not mistaken, Maxwell's equations were originally written in cumbersome quaternion notation! However, because quaternions are good in describing rotations in $\text{3D}$ and do not suffer from problems as the gimbal lock, they are still used today in computer simulations, along with Euler angles and matrices.

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