Tag Archives: William Rowan Hamilton

Imaginary Numbers: From Outcast to Respectability

Image: Matheepan Panchalingam, via Flickr.

Image: Matheepan Panchalingam, via Flickr.

Imaginary numbers, which are also known as complex numbers, have had a pretty bad reputation. When most people think of imaginary numbers, they probably break out in a cold sweat from the horrific memories of high school math class. They think that imaginary numbers are utterly incomprehensible and useless in the “real” world. “Imaginary numbers” sound very intimidating to people who are not familiar with them. They also sound highly theoretical with little or no use outside of pure mathematics. In fact, the exact opposite is true.

The most common imaginary number is i, which is formally defined as i = √-1. Since the act of squaring any real number always makes the number positive– whether it began as a negative number or not, it is impossible to find the square root of a negative number without using i. Thus, i made possible an entire class of math problems that were not possible before. For example, √-64 = 8i, cannot be done without using i, because √-64 does not exist in the real number line. Additionally, i can be easily changed from an “imaginary” number into a “real” number simply by squaring it: i² = -1.

The first known person to stumble upon the idea of using an imaginary number to take the square root of a negative number was the Greek mathematician Heron of Alexandria in 50 CE. He was trying to find the volume of a section of a pyramid using a formula that involved the slant height of the pyramid. However, certain values for the slant height would produce the square root of a negative number. Heron was very uncomfortable with this result, so in order to avoid using a negative number, he fudged his calculation by dropping the negative sign.

Girolamo Cardano was an Italian mathematician who was particularly interested in finding the solutions to cubic and quartic equations. In 1545, he published a book titled Ars Magna, which contained the solutions to cubic and quartic equations. One of the equations in his book gave the solution of 5 ± √-15. Commenting on this equation, Cardano wrote, “Dismissing mental tortures, and multiplying 5 + √ – 15 by 5 – √-15, we obtain 25 – (-15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless.”

Perhaps the first champion of imaginary numbers was Italian mathematician, Rafael Bombelli (1526-1572). Bombelli understood thattimes should equal -1, and that -i times should equal one. However, Bombelli could not find a practical use for this property, so he generally was not believed. Bombelli did have what people called a “wild idea” – that imaginary numbers could be used to get real answers.

Imaginary numbers continued to live in disgrace until the work of a series of mathematicians in the 18th and 19th centuries. Leonhard Euler helped clear up some of the problems with using imaginary numbers by developing the notation i to mean √-1. He also introduced the notation a+bi for complex numbers. Carl Friedrich Gauss  made imaginary numbers much more concrete and less “imaginary” when he graphed imaginary numbers as points on the complex plane in 1799. However, William Rowan Hamilton in 1833, delivered the coup de grace to imaginary numbers’ bad name when he advanced the idea that complex numbers could be expressed as a pair of real numbers. For example 4+3i could be written simply as (4,3). This made complex numbers much easier to understand and use.

Today, imaginary numbers are an essential part of the everyday calculations that make modern technology work. They are indispensable in the field of electrical engineering, particularly in the analysis of alternating current, like the electrical current that powers household appliances. Also, cell phones and air travel would not be possible without imaginary numbers because they are necessary in the computations involved in signal processing and radar. Imaginary numbers are even used by biologists when studying the firing events of neurons in the brain. Imaginary numbers have come a long way in the five hundred years since they were scoffed at for being absurd and totally useless.

Sources:

http://nrich.maths.org/5961

http://www-history.mcs.st-andrews.ac.uk/Biographies/Cardan.html

https://www.flickr.com/photos/mpancha/2505656136/in/photolist-

http://plus.maths.org/content/imaginary-tale

http://rkbookreviews.wordpress.com/2010/01/10/imaginary-tale-summary/

http://rossroessler.tripod.com/

http://mathforum.org/library/drmath/view/53879.html

https://www.math.toronto.edu/mathnet/questionCorner/complexorigin.html

https://www.flickr.com/photos/mpancha/2505656136/in/photolist-

Vandalism and Mathematics

PROTIP: You can get around the Shannon-Hartley limit by setting your font size to 0.

Image: Randall Munroe.

Regarding the comic

This xkcd comic has two points.  The first is understandable without any context. If the writer had in fact discovered a proof that information is infinitely compressible, then ANY amount of space would be sufficient to contain it. The second point refers to Fermat’s famous statement “I have discovered a truly remarkable proof of this theorem which this margin is too small to contain,” which was, of course, referring to Fermat’s Last Theorem, a topic which we discussed extensively in class.

Liberal use of others property

It is now often believed that Fermat did not actually have a correct proof of this theorem. This minor detail did not, however, deter the great Fermat from writing it as fact in the margin of his copy Arithmetica to be discovered posthumously and baffle mathematicians for centuries to come. This, however, is not the only case of mathematicians writing statements in strange places. Another mathematician who did this was William Rowan Hamilton. Unlike Fermat, Hamilton decided to actually carve in his answer to a question, as opposed to carving in a claim that he has an answer. To be fair though, Fermat did own his book, while Hamilton didn’t actually own the bridge. This occurred in 1843, while taking a walk, he had a flash of brilliance during which he discovered Quaternions. Lacking a proper way with which to write down the result, Hamilton instead chose to carve his answer in the side of a bridge.

But what are quaternions?

Hamilton knew how to add and multiply complex numbers in a plane. However, he did not know how to multiply them in space. Quaternions were his solution to this problem, because while he could not figure out how to multiply complex points in a 3-dimensional space, he could figure out how to do it in a 4-dimensional space. In fact there is now a theorem which says the only normed division algebras which are number systems where we can add, subtract, multiply, and divide, and which have a norm satisfying |zw|=|z||w| have dimension 1, 2, 4, or 8. Quaternions can be thought of as a 4-dimensional space and are often denoted by H or ℍ. They are a noncommutative number system over the complex space, which just means that a*b does not necessarily equal b*a. They are defined as ℍ ={a+bi+cj+dk} where a, b, c, and d all belong to the real numbers. Note in particular that ij = k = -ji, jk = i = -kj, and ki = j = -jk. This eventually leads to what Hamilton engraved on the Brougham Bridge: i2 = j2 = k2 = ijk = -1, which means that ij, and are all equal to square root of -1.

Utility of quaternions

The quaternions can be used to do rotations in 3 dimensions, which may seem unintuitive given that quaternions describe a 4-dimensional space. To better explain this we need the concept of real and pure quaternions. A real quaternion is one which contains only a real part, while a pure quaternion is one which does not contain a real part. This is the equivalent of partioning a complex number into its real and imaginary parts. The difference between these two scenarios is that the pure portion of a quaternion is a vector in 3-space instead of a single number. Thus a real quaternion will take the form [a, 0] where 0 is the zero vector and a pure quaternion will take the form [0, v] where v is a vector of the form v=bi+cj+dj. Note that this means that the set of all pure quaternions define a 3-space. Thus the process of rotating in three dimensions is accomplished by starting with a pure quaternion, called p. This quaternion is then multiplied by the rotor, a second quaternion, called q, of the form [cos(Θ), sin(Θ)*v] where v  is a vector of the form v=bi+cj+dj and Θ is the angle by which we are rotating. If p happens to be perpendicular to q then the result will be a pure quaternion and the process is complete. However, if it is not the resulting quaternion will not be pure and the magnitude will be off. We can, however, multiply this new result by the inverse of q which will result in a pure quaternion of the desired length. Note that this means that the object should start and end in the 3-dimensional space as defined by the set of all pure quaternions with the real portion being used as an intermediary. I should also mention that the inverse is the conjugate of the quaternion divide by its normalization squared, where the conjugate is computed by negating the vector v and the normalization by dividing by the magnitude of the quaternion. Quaterions, however, don’t just allow for rotation in 3 dimensions, they also help avoid certain problems such as gimbal lock. Gimbal lock occurs when two out of the three rotational axes align. When this happens, the aligned axes both rotate the object in the same way. While you can still get out of the gimbal lock, it does force you to do some additional rotations. Quaternions circumvent this problem by having that intermediary 4th rotational axis.

Conclusion

If you want to commit vandalism, all you have to do is discover something brilliant which will be used for quite some time after its discovery in technologies which have yet to exist and engrave it in the side of a bridge or scribble it within the margins of a book. You might even get a plaque commemorating your vandalism.

Image: JP, via Wikimedia Commons.

Sources

http://math.ucr.edu/~huerta/introquaternions.pdf

http://www.cprogramming.com/tutorial/3d/quaternions.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

http://en.wikipedia.org/wiki/Quaternion

http://www.gamedev.net/page/resources/_/technical/math-and-physics/do-we-really-need-quaternions-r1199

http://www.gamedev.net/topic/25314-why-diana-grubers-wrong-about-quats/

http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://3dgep.com/understanding-quaternions/

https://www.youtube.com/watch?v=zc8b2Jo7mno

A History of Mathematics, Uta C. Merzbach and Boyer.