6 Roots Of Unity
This blog is called 'Roots of Unity' because in 2004 I thought it would be an awesome band name.
By a primitive nth root of unity we mean a solution?of z? = 1 which has the property that its powers are exactly the solutions of this equation in C. For example is a primitive nth root of unity. (a) Find all primitive 6th roots of unity. (b) Find all primitive 5th roots of unity. (c) For which values of k, 0 a primitive nth root of unity? By the method of subsection 5.1, we compute all the roots of unity: For, we have the following roots: The images in Gauss-Argand plane of the roots of unity are the vertices of a regular polygone inscribed in the unit circle. Dec 16, 2017 There are infinitely many roots of unity, densely occupying the unit circle in the complex plane, but we don’t want our emblem to be just a circle, so let’s call special attention to the simplest roots of unity: say the 1st, 2nd, and 3rd roots of unity.
Not the cover of my band's first album. Image courtesy of Gregory A. Moore.
I encountered the term in a college math class, and I had no illusions that I would be in a band at any time in the near future, but it seemed prudent to have some good band names in my back pocket, just in case. You never know when an awesome band will be looking for a recreational violist who already has a nerdy band name picked out.
So what is a root of unity anyway? The word 'root' in the term refers to square roots, cube roots, and any other roots you might happen to need. For any integer n, the nth root of a number k is a number that, when multiplied by itself n times, yields k. The word 'unity,' perhaps a bit anticlimactically, just means 'one.' So a root of unity is any number which, when multiplied by itself some number of times, yields 1.
At this point, the definition really seems like it's making a mountain out of a molehill: 1 and -1 are the only numbers that seem to fit the bill. But the evocative phrase doesn't usually come up when we're talking about real numbers: we need to work with complex numbers to get any interesting roots of unity.
The complex numbers consist of all numbers that can be written in the form a+bi, where a and b are real numbers and i is the square root of -1. The number i doesn't exist on the real number line because any real number multiplied by itself is positive, so the letter i, standing for 'imaginary,' is used instead. Imaginary numbers aren't any more imaginary than the numbers 6, 8, or 27,000, but the label has stuck. The number i itself is a root of unity: i2=-1, so i4=1, making i a 4th root of unity. Any square, cube, or other roots of i are also roots of unity.
6 Roots Of Unity Lutheran Church
To see what makes roots of unity special, we need to delve a little bit into notation. If you don't like notation, you should probably skip the next three paragraphs.
We can think of the set of complex numbers as a 2-dimensional plane. We specify a complex number with two coordinates the same way we would on a graph: the point (1,1) refers to the number 1+i. If we were standing on the point (0,0), we probably wouldn't want to walk one unit over and then one unit up to get to the point (1,1). Instead, we'd take a diagonal by walking √2 units at an angle of 45 degrees to the x-axis. Notationally, when we use this radial distance and direction, we write a complex number as rei θ, where r is the distance and θ is the direction, usually written in radians rather than degrees. There are 2π radians in a whole circle, so the number 1 is written ei 2π. The angle 45 degrees is π/4 radians, so the point (1,1) that we found above would be written as √2ei π/4.
The complex number 1+i, written in rectangular and polar coordinates.
This method of specifying a point with a length and a direction is called using polar coordinates, and it shows up most beautifully in Euler's identity, eiπ=-1.
Polar coordinates make it really easy to multiply complex numbers. With a+bi notation, you have to FOIL (remember middle school algebra?), and you end up with a total of four terms you have to add together. But with the polar notation of reiθ, it's really easy: you multiply the distances together and add the angles. So 5eiπ/6× 2eiπ/3=10 ei π/2. So multiplying involves both expansion or contraction (that's the distance part) and rotation (that's the angle part).
I think the most beautiful thing about roots of unity (besides the awesome name) is that they are kind of a balancing point between 0 and infinity. What I mean by that is if we have a number written rei θ which, when multiplied by itself a certain number of times, yields 1, the distance r itself must be 1. If r were larger than 1, say 2, then as we multiplied the number by itself more and more times, its distance from (0,0) would go from 2 to 4 to 8 and on and on, spiraling out to infinity. If r were smaller than 1, say 1/2, the point would spiral in to 0: 1/2, 1/4, 1/8, and so on. 1 is the only radial distance that will stay perfectly balanced, just marching around the circle as we multiply the numbers together.
To make this more concrete, I happen to know that ei2π/7 is a root of unity. When I raise it to the 7th power, I get ei2π, which is 1. Each time I multiply ei2π/7 by itself, I rotate 1/7 of the way around the circle. In fact, as I multiply ei2π/7 by itself successively, I get this picture, my blog banner.
The 7th roots of unity. Image courtesy of Gregory A. Moore.
In fact, there are seven 7th roots of unity, and each gold disc in that picture is one of them. We can get an nth root of unity for any number n by replacing the 7 in ei 2π/7 by n. The pictures for other roots of unity look a lot like that diagram above, they just have a different number of gold discs.
I'm not sure the mathematical definition of a root of unity stands up to the awesomeness of the phrase as a band name, but I do think it's a beautiful idea, and it's very useful in complex analysis. I'm still collecting awesome band names now that Roots of Unity is out. Two current frontrunners are 'The Butterfly Assumption,' which is related to a friend's area of math research, and 'Premeditated Rosemary Theft,' which is just related to an unfortunate incident involving my balcony and some missing herbs.
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THE ROOTS OF UNITY
Negative and imaginary numbers
I will now venture into another discipline, algebra. The use of negative and imaginary numbers in algebra confirms the fourfold nature of analysis, and also provides additional insight into the threefold. Here again, I should acknowledge that I am using the concepts of a discipline for purposes outside their usual application, but there are certain discoveries of mathematics which make valuable contributions to our study.
The evolution of mathematics was given great impetus by the discovery that it was possible to use negative numbers, negative quantities. If we represent positive numbers extending to the right of zero, we can represent negative numbers extending to the left:
etc. . . . -3 -2 -1 0 +1 +2 +3 . . . etc.
With this device, we may describe addition as moving to the right, and subtraction as moving to the left. This makes it possible to subtract a larger number from a smaller one; for instance, if we take 3 from 1, we get -2, which is a real (although negative) quantity.
6 Roots Of Unity Church
Another important concept was that of imaginary numbers. They were not so much discovered as encountered.
Mathematics had arrived at the concept of a number as having roots; numbers which, multiplied together, will produce that number. When the concept of negative numbers came along, there was a clash. What would be the two numbers which multiplied together would produce a negative quantity, -1, for example? For a time there was no answer. The square root of a negative quantity must be impossible. So it was called imaginary. But when Gauss, called by Bell the prince of mathematicians, found a method for representing imaginary numbers, it was not long before their value was appreciated, and today they are just as important as real numbers. Thismethod uses the Argand diagram, which, in essence, correlates unity to the circle, and roots of unity to fractions of the circle.
Recall that negative numbers were pictured as extending ina direction opposite to positive numbers. In this way, the square roots of unity, +1 and -1, can be expressed as the opposite ends of a line with center zero. This line can be thought of as an angle of 180 degrees, or a diameter.
Gauss extended the idea further and pictured as halfway between
+ 1 and - 1,or an angle of 90 degrees from the line -1 and +1. Thus, if the division of unity into plus and minus is a diameter, or 180 degrees, a second division leads to an axis which 'mediates' this diameter, or an angle of 90 degrees.
Thus we have two axes - the horizontal representing positive and negative real numbers, and the vertical representing positive and negative imaginary numbers. These two axes form the complex coordinate system, and a number on the planedescribed by these axes is a number having a real part and an imaginary part.
Using the Argand diagram, this circle of unit radius (radius = 1) on the complex coordinate system, the other roots of unity (cube roots, fifth roots, etc.) are found simply by dividing the circle into three, five, etc., equal parts. Finding the roots of unity becomes simply a matter of inscribing polygons within the unit circle: a triangle for cube roots, a pentagon for fifth roots, etc. The roots are the points on the circle; their values have a real part and an imaginary part, and are measured along the horizontal and vertical coordinates respectively. This means that they are measured in terms of square roots and fourth roots.
6 Roots Of Unity Tree
From this extremely powerful simplification, it follows that all analysis is fourfold - any situation can be analyzed in terms of four factors or aspects. This not only confirms Aristotle (his four causes) but explains why quadratic (literally, 'four-sided')equations occur so frequently in mathematics.
But the important generalization that all analysis is fourfold works both ways. It shows both the extent of the fourfold and the limitations of analysis, for there are things in the content of experience that are beyond analysis.
Staying within the geometrical method already set up, we can show that these non-analytic factors involve three-ness, five-ness, and seven-ness. Despite the fact they can be described analytically, this description fails to capture their true nature.
Cube roots and the threefold operator
Nth Roots Of Unity
As I said, we can express the cube roots of unity analytically in terms of square roots; that is, using the two-dimensional diagram.
These roots are obtained by inscribing an equilateral triangle within the circle. One of these roots is +1; the other two are points half a unit to the left of the vertical diameter, and half a side (of the triangle) above and below the horizontal diameter. Since the side of the triangle is ,
the vertical coordinates are
the i being used to indicate that we are measuring in the vertical direction. (The imaginary is commonly represented by i.)
By this we see that the values for the cube roots of unity can be expressed as square roots.
But is an irrational number, meaning that it is not a ratio of whole
numbers.Since is the diagonal of a unit square,* we might expect to
find some expression for in units. We need not look far to find what
this is, for is the diagonal of the unit cube.**
* A unit square is a square whose side is 1.
**A unit cube is a cube whose side is 1.
So in order to represent in unitary fashion,
we must leave thetwo-dimensional plane. Full representation of the cube roots of unity ultimately involves the three dimensions of space. The threefold operator, represented analytically as equidistant points on the circle, is actually a three-dimensional activity, whose measure gives only its analytic aspect. The analytic aspect, which is in two dimensions, does not convey the full meaning of the cube root; it is like the shadow of a solid figure.
The threefold nature of the cube root is nonanalytic. It involves categories which differ from one another more profoundly than those of the fourfold.
Here we have a formal device to show the inadequacy of analysis for a complete account of the world. This is but one example of the fundamental distinction between the threefold and fourfold operators, a distinction so important to our theory and to life that I will devote the next chapter to a comparison between these two operators.