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The formula \( (a+b)^2=a^2+2\cdot a\cdot b+b^2 \) leads to the binomial coefficient \(\binom{n}{k}=\frac{n!}{k!\cdot\left(n-k\right)!}\) in pascal's triangle and to the binomial theorem \(\left(a+b\right)^n = \sum\limits_{k=0}^{n}\binom{n}{k}\cdot a^{n-k}\cdot b^{k}\). In part II thale's theorem is derived and leads with the binomial theorem to the pythagorean theorem \( c^2=a^2+b^2 \).

The formula \( (a+b)^2=a^2+2\cdot a\cdot b+b^2 \) leads to the binomial coefficient \(\binom{n}{k}=\frac{n!}{k!\cdot\left(n-k\right)!}\) in pascal's triangle and to the binomial theorem \(\left(a+b\right)^n = \sum\limits_{k=0}^{n}\binom{n}{k}\cdot a^{n-k}\cdot b^{k}\). In part II thale's theorem is derived and leads with the binomial theorem to the pythagorean theorem \( c^2=a^2+b^2 \).

Several means are visualized: arithmetic mean, geometric mean and the harmonic mean.
The more complex concepts of root mean square and variance follow the means and are displayed in figures.

Fractions and percent are compared and are followed by the concept of gradual and continuous growth, which leads to euler's number e.
In part II the concept of euler's number e is expanded to the natural exponential function and compared to exponential growth

The basic types of calculation with fractions are visualized: addition, subtraction, multiplication and divsion.
The needed concepts therefore: reduction, expansion and the reciprocal fraction are explained.

The circumference of a circle is derived step by step, which leads to the number π.
The derived insights are used to determine the circumference and area of any size of a circle.

The addition and subtraction theorems are visualized.
Basic values for the sine and cosine functions are derived, which lead together with the sum theorems to the general sine, cosine and tangent functions.

The unit circle is introduced and the difference between the units degree and radiant is explained.
Along with the unit circle the trionometric formulas are presented.
Further useful insights are gained and finally the sine and cosine theorem are derived.

The babylonian root is presented and extended to the nth root algorithm.
The way the geometric- and arithmetic means are involved also leads
to the statement: geometric mean ≤ arithmetic mean.

A formula for the sine value for the half of an angle is visulized.
Its used to derive the arcus sine, arcus cosine & arcus tangent functions.

The pq-formula is introduced to solve a quadratic equation.
The problem with negative roots leads to the imaginary number i and to complex numbers.
Part II introduces the calculation with complex numbers.
In part III the purpose of complex numbers is visualized along with 2 new formulas.

Euler's formula is derived, which also leads to new trigonometric functions.
The special case of euler's identity is visualized.
Finally new complex forms are introduced and all four forms of complex numbers are compared.