User:Numiri/KinderCalculus/bionic numbers

Bionic numbers is the KinderCalculus name for Complex numbers. With children, naming a lesson "Complex" and to study "Imaginary" objects is tantamount to setting the teaching exercise up for failure. We choose the term Bionic because the numbers have 2 components and they are more powerful -- that is, they can solve more equations. Bionics # have 2 ticks: ( horizontal, vertical ). We draw them on a number sheet, instead of a number line.

Here's the KinderCalculus vocabulary for bionic numbers:   tick = coordinate  grid = coordinate system  round ticks = polar coordinate  square ticks = cartesian coordinate 

Motivation
There are lots of equations we cannot solve with reals, x^2 = -1, so we need bionic. There are no numbers on the line that will fit this self-times. We need something else.

Let's go back to Kindergarten. We started out with the counting numbers -- 1, 2, 3, ... With them, we can state how much money we have -- $1, $2, $3, ...  But we cannot state how much money we owe. The negatives are missing. But it can be argued that having money and owing money are completely different things, so the counting numbers can be forgiven for not covering 2 different concepts.

Worse yet, if we use the counting numbers to indicate distances from where we stand, we can only use distances in one direction, say 10ft to the right of us. Those numbers cannot point to the left. Now, the left side of our body and the right side of our body are equally valuable, unlike the example of having money and owing money, so we cannot forgive the counting numbers for its gaps. Counting numbers are lop-sided. We even it out with negatives.

The same can be said of the real numbers -- they are lopsided. If positive and negative numbers are on par like the left and right side of your body, then why is it that we cannot draw a negative snowflake for x^2 = -8, but we can for x^2 = 8? Bionic numbers brings justice to the left side of the number line.

The rest of bionic number arithmetic follows the normal curriculum.

Bionic Arithmetic
Bionic arithmetic is best visualized by the "stick" operation below: ( a    b )

|    |          |     |  |     |     =    | ,  |           =  ( a+c , b+d ) = (a+c) + i(b+d) |    |          |     |

( c    d )

( a     b )

| \ / |         |   |     \ /  |  x   |    =    | - | ,   +     =  ( ac - bd, ad + cb ) = (ac - bd) + i(ad + cb) | / \ |         |   |     / \

( c     d )

Visualizing Bionic Arithmetic
Bionic number arithmetic is still much the same as real number line arithmetic:
 * bionic addition is vector addition, which is a shift in a vector on the number line
 * bionic multiplication, multiplying size and rotating angles also have origins in real number line multiplication, with the angle on the real number line being 180 degrees.

Below is a convenient depiction of bionic number addition. When we add bionic numbers, the shift of vector A is analogous of a shift of a length in adding real numbers along the number line.

The Thumbprint of God
With bionic numbers, we now have an accessible introduction to how the world might have been formed, via Mandelbroit fractal set. Fractals offer one glimpse of how self-similarity may have formed the intricate patterns in nature -- from leaves, to trees, to mountains. For this reason fractals are sometimes referred to as the thumbprint of god. God here refers to Nature, not necessarily any religious diety. Fractals are the signature in these patterns, much like a thumbprint signature.

Special Relativity
A variant of bionic numbers, the Split Complex numbers offer a more compact form of the Lorentz Transformation via hyperbolic functions cosh & sinh ...

Exercises
Visualizing bionic arithmetic:

assign the standard exercises for adding, subtracting, multiplying a bunch of bionic numbers ...