This article may be too technical for most readers to understand. (August 2015) |
Distance
The task is to define the distance of a point in R^{2}. As you will see later, it's more desirable to define the distance of each complex number first then define the distance of each point in R^{2} to be the same as the distance of the complex number whose real part is the first coordinate of that point and whose imaginary part is the second coordinate of that point. For any complex number c, a left complex multiplication by c is the operation from C to C that assigns to each complex number z, c × z. A left complex multiplication is an operation such that there exists a complex number c that it's a left complex multiplication by. A distance function is function from R^{2} to R such that the following conditions hold:
- The distance of every nonnegative real number r is r.
- The distance of any complex number is a nonnegative real number.
- The distance of any complex number is equal to the distance of its complex conjugate.
- For any 2 complex numbers, the distance of their product is equal to the product of their distances.
Suppose d is a distance funtion, then for any complex number a + bi, (d(a + bi))^{2} = d(a + bi)d(a + bi) = d(a + bi)d(a - bi) = d((a + bi)(a - bi)) = d(a^{2} + b^{2}) = a^{2} + b^{2} so d(a + bi) = √a^{2} + b^{2}. Since only d(x, y) = √x^{2} + y^{2} can be a distance function, we will define the distance of a point to be the number that function assigns to that point. In fact, d(x, y) = √x^{2} + y^{2} satisfies the conditions of a distance function because:
- For any nonnegative real number r, d(r) = d(r + 0i) = √r^{2} + 0^{2} = √r^{2} = r
- For any real numbers a, b, a^{2} and b^{2} are both nonnegative real numbers so a^{2} + b^{2} is also a nonnegative real number so √a^{2} + b^{2} which is equal to d(a + bi) is a nonnegative real number as well.
- d(a + bi) = √a^{2} + b^{2} = √a^{2} + (-b)^{2} = d(a + (-b)i) = d(a - bi)
- d((a + bi)(c + di)) = d((ac - bd) + (ad + bc)i) = √(ac - bd)^{2} + (ad + bc)^{2} = √a^{2}c^{2} - 2abcd + b^{2}d^{2} + a^{2}d^{2} + 2abcd + b^{2}c^{2} = √a^{2}c^{2} + a^{2}d^{2} + b^{2}c^{2} + b^{2}d^{2} = √(a^{2} + b^{2})(c^{2} + d^{2}) = √(a^{2} + b^{2})√(c^{2} + d^{2}) = d(a + bi)d(c + di).
For any complex numbers c and d, the distance from c to d is the distance of d - c. On the other hand, it can't be proven that there is a way to define volume for all subsets of R^{3} such that the intuitive properties of volume hold.
Rotation
The unit circle is the set of all points with a distance of 1. An origin rotation is a left-multiplication by any complex number on the unit circle, that is, any complex number a + bi where a^{2} + b^{2} = 1. A transformation T from C to C is a rotation about c iff adding c then applying T then subtracting c results in an origin rotation. A rotation is any operation for which there exists a complex number c that it's a rotation about. It turns out that any 2 origin rotations commute because complex multiplication is commutative and the composition of any 2 origin rotations is an origin rotation because complex multiplication is associative. It is not however true that the composition of any 2 rotations is a rotation. The mistake that it is true comes from the following incorrect argument:
- For any 2 rotations, for each rotation, there exists a point that it is a rotation about so there exists a point that both rotations are a rotation about so their composition is a rotation about that point so their composition is a rotation.
Unit circle
Some people might alternatively define the unit circle to be the set of all points on the trajectory of a point particle that is at the complex number 1 at time 0 and whose derivative with respect to time (velocity) is always equal to its position × i. Let z(t) be the position of the particle at time t for any real number t. For any real number t, cos(t) is defined to be the real component of z(t) and sin(t) is defined to be the imaginary component of z(t). There must be some justification for defining the unit circle that way, that is, that for any real number t, (cos(t))^{2} + (sin(t))^{2} = 1. Indeed, it can be proven that for all t, (cos(t))^{2} + (sin(t))^{2} = 1 as follows:
- Each of the 2 complex differential equations of z(t)
- z(0) = 1
- z'(t) = i(z(t))
- can be split into 2 real differential equations giving a total of 4 differential equations:
- cos(0) = 1
- sin(0) = 0
- cos'(t) = -sin(t)
- sin'(t) = cos(t)
- so ddt((cos(t))^{2} + (sin(t))^{2}) = ddt((cos(t))^{2}) + ddt((sin(t))^{2}) = 2(cos(t))(-(sin(t))) + 2(sin(t))(cos(t)) = -2(sin(t))(cos(t)) + 2(sin(t))(cos(t)) = 0, so (cos(t))^{2} + (sin(t))^{2} is constant and so is always 1 so (cos(t)) + (sin(t))i is on the unit circle for every real number t so the particle's trajectory is entirely on the unit circle.
In addition to that, from z'(t) = i(z(t)), we can derive that z(t) is an exponential function and that z(-t) is the complex conjugate of t. That is,
- z(s + t) = z(s)(z(t))
- z(-t) = z(t)
Converting this into real and imaginary parts, we get
- cos(s + t) = cos(s)cos(t) - sin(s)sin(t)
- sin(s + t) = sin(s)cos(t) + cos(s)sin(t)
- cos(-t) = cos(t)
- sin(-t) = -sin(t)
Rotating the unit square doesn't change its area
Proof:
Here's a Reader proof. The unit square is the set of all ordered pairs of real numbers such that both coordinates are in the closed interval [0, 1]. For any integrable subset of R^2, its area is defined by integrating with respect to 1 coordinate than integrating with respect to the other coordinate. The area of some nonintegrable sets including a rotated square can also be defined by integrating a ralated function that doesn't assign any value to the points on its edge. For any unit square rotated by a complex number with both parts positive with the real part larger than the imaginary part, let a be the real part and b be the imaginary part of that number. That square can be split into 4 right angle traingles with legs of length a and b and a square with edges of length a - b. Then its area is 4(ab2 + (a - b)^{2} = 2ab + (a^{2} - 2ab + b^{2}) = a^{2} + b^{2} but a^{2} + b^{2} = 1 so the area of the square is 1.
Complete proofs
From Finite cardinality subtracting theorem, it can be shown that there exists a variation of New Foundations whose formal proof system is consistent and only describes statements about number theory and allows you to prove certain describable statements that are provable only by converting that statement into a statement about number theory then writing a formal proof of it. Some of those statements include the statement that there is exactly one distance function which can be proven by first proving that the set of all real numbers with the addition, multiplication, and inequality relation satisfy the axioms described at https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Synthetic_approach. The set of all real numbers is constructed from the set of all natural numbers. The statement that that variation of New Foundations is that type of theory can also be proven by converting it into a statement about number theory then writing a formal proof of it in a higher level formal proof system about number theory. A savant with certain desirable traits would be able to figure out how to write a complete formal proof of any of those statements described earlier in that variation of New Foundations and not a reader proof or intuitive proof. For example, the proof of the Pythagoren by similar traingles shown at https://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_using_similar_triangles and the proof by rearrangement shown at https://commons.wikimedia.org/wiki/File:Pythag_anim.gif are intuitive proofs, not formal proofs.
Usage
Distance is a powerful tool for proving so many geometry theorems. Some schools in the future might decide to teach students what a distance function then announce that they're going to prove there is exactly one distance function, and do so by teaching the proof that only the function that assigns √a^{2} + b^{2} to a + bi for all a and b can be a distance function, then teaching that it has already been proven that that function is a distance function without teaching the proof, so they will accept students accepting that the 4 conditions are true without a proof and using it to prove other theorems. For example, a student might be asked to prove on a test from those 4 conditions that for all real numbers x, y, d(x + yi) = √x^{2} + y^{2} where the following answer would be worth full marks:
- d(a + bi) = √(d(a + bi))^{2} = √d(a + bi)d(a + bi) = √d(a + bi)d(a - bi) = √d((a + bi)(a - bi)) = √d(a^{2} + b^{2}) = √a^{2} + b^{2}
and then that test will be taken up later with the teacher explaining that that proof shows that distance could not have been defined another way. When that student is asked to prove on a later test that for all real numbers x, y, xy = (x + y)^{2} - (x - y)^{2}4, the teacher will accept it if the student proves it as follows:
- xy = 2xy2 = (x^{2} + 2xy + y^{2}) - (x^{2} + y^{2}2 = (x + y)^{2} - (x^{2} + y^{2})2 = (x + y)^{2} - (d(x + yi))^{2}2 = (x + y)^{2} - (d((0.5 - 0.5i)((x - y) + (x + y)i)))^{2}2 = (x + y)^{2} - 0.5(d((x - y) + (x + y)i))^{2}2 = (x + y)^{2} - 0.5((x - y)^{2} + (x + y)^{2})2 = 0.5(x + y)^{2} - 0.5(x - y)^{2}2 = (x + y)^{2} - (x - y)^{2}4.
In a later course, the teacher might decide that since it's already been proven that d(cos(t) + (sin(t))i) = 1 for all real numbers t, they will accept students knowing that it has been proven without ever learning or understanding the proof and proving other theorems on a test from it.
Alternate definition
Some people might define the distance formula to be the function our universe uses, that is, the function that assigns to each point on a plane, the y-coordinate of the point on the positive y axis a compass can move that point to when its needle is at the origin. There's a really good proof physical proof that that function is a distance function, which is seeing a rigid object rotate with your own eyes. It turns out that the universe had to pick that function and couldn't have picked another one because it's the only distance function. It can be mathematically proven that that function is in fact √a^{2} + b^{2} as was already shown earlier. With a physical proof that an origin rotation, a left multiplication by a complex number with a distance of 1, preserves the distance of all points, not just those on the positive x-axis, some people might be fine with doing away with a mathematical proof.
Rational number system
We can't actually prove irrational numbers exist. If the plane consists only of points in Q^{2}, there is no function that assigns to each point in Q^{2} a rational number that satisfies the properties of a distance function because there is no number whose square is 2.