Number circle (no ontology)

From Hi.gher. Space

The number circle, similar to the number line, is a mathematical device used to represent numbers.

Formation

The number circle is formed by drawing a circle of radius 0.5 units above the number line such that the bottommost point on the circle is at zero on the line. Then, by drawing a line l connecting the top of the circle to any point on the number line, the number on the circle where l intersects it (other than the top) is equal to the number on the number line where l intersects it. Thus, zero is at the bottom, one is at the right, negative one is at the left, and the top is defined as infinity as it is the limit of the numbers to its sides.

Key features

The biggest advantage of using the number circle rather than the number line is that the circle includes infinity. However, the disadvantage is that greater than and less than become meaningless.

When using the number circle, positive and negative infinity are exactly the same number, just as positive and negative zero are.

Taking a point on the number circle and reflecting it in the horizontal diameter of the circle will give the reciprocal of that number. Similarly, reflecting in the vertical diameter will give the negative of that number. It can be derived, therefore, that if two points on the number circle represent the gradients of two perpendicular lines, the line connecting the points is a diameter of the circle.

Derivation of number positions

Assuming that the number circle is centered at the origin, the equation for the circle is:

x² + y² = ¼

The equation of the line is:

y = ½ - x/m

To find the x-position of the point on the number circle we can solve the equations simultaneously:

x² + (½ - x/m)² = ¼
∴ x² + ¼ - x/m + x²/m² = ¼
∴ x² - x/m + x²/m² = 0
∴ x²m² - xm + x² = 0
∴ x(x(1+m^-2)-m^-1) = 0
∴ x(1+m^-2)-m^-1 = 0 [or x = 0, but this solution is unwanted]
∴ x = m^-1/(1+m^-2)
∴ x = m/(m²+1)

To find the y-position of the point is a simple case of substitution:

y = ½ - x/m
∴ y = ½ - (m²+1)^-1

To find the angle of the point (x,y) around the number circle (anticlockwise from the point 1), we use trig:

θ = tan^-1(y/x)
∴ θ = tan^-1((½ - (m²+1)^-1)/(m/(m²+1)))
∴ θ = tan^-1((m² - 1)/2m)

Finding m

It is possible to find m given θ.

θ = tan^-1((m² - 1)/2m)
∴ tan(θ) = (m² - 1)/2m
∴ 2mtan(θ) = m² - 1
∴ 0 = m² - 2mtan(θ) - 1
∴ m = (2tan(θ) ± √(4tan(θ)² + 4))/2
∴ m = tan(θ) ± √(tan(θ)² + 1)

The negative solution is wanted if θ is between π/2 and 3π/2; the positive solution otherwise.

The formula can be rewritten as m = tan(θ/2+π/2); this is proven as follows:

Prove that tan(θ) + √(tan(θ)² + 1) = tan(θ/2+π/2)
Let t = tan(θ/2)
LHS = sin(θ)/cos(θ) + 1/cos(θ)
= (1+sin(θ))/cos(θ)
= (1+2t/(1+t²))/((1-t²)/(1+t²))
= (1+2t+t²)/(1-t²)
= (1+t)²/(1+t)(1-t)
= (1+t)/(1-t)
= tan(θ/2+π/2)
= RHS

Connection with the trigonometrical tangent

Graphing y = tanx on a cylindrical graph where y is circular and x linear produces an infinite, un-warped helix. This has been confirmed with a Visual Basic program, for which the source code will be available for download soon. It can also be mathematically proved as follows:

θ = tan^-1((m² - 1)/2m) [the above result]

If θ is the angle resulting from projecting tan(x) onto the number circle, i.e. m = tan(x), substitution gives:

θ = tan^-1((tan(x)² - 1)/2tan(x))

The double angle formulae state that tan(2x) = 2tan(x)/(1 - tan(x)²), so:

(tan(x)²-1)/2tan(x) = -tan(2x)^-1
∴ θ = tan^-1(-tan(2x)^-1)

Now tan(2x-π/2) = sin(2x-π/2) / cos(2x-π/2)
= (sin(2x)cos(π/2)-cos(2x)sin(π/2)) / (cos(2x)cos(π/2)+sin(2x)sin(π/2))
= -cos(2x)/sin(2x)
= -tan(2x)^-1
∴ θ = tan^-1(tan(2x-π/2)) = 2x-π/2

Since θ now varies linearly in relation to x, the graph must be a helix.

Extension to the complex numbers

A number sphere can be formed from the complex plane in exactly the same way as the number circle is formed. Such a sphere would have zero at the bottom, infinity at the top, one at the right, negative one at the left, i at the "back", negative i at the "front" (assuming the top of the complex plane was assumed to be the back and the bottom the front). The distance up the sphere would represent the absolute value of the number.

In the number sphere, there is also no such thing as a directed infinity: there is only one infinity, and it does not have a defined angle, just as zero does not have a defined angle.