Due to the nature of the mathematics on this site it is best views in landscape mode.

It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: Euler's formula describes two equivalent ways to move in a circle. This stunning equation is about spinning around?

Yes -- and we can understand it by building on a few analogies: Starting at the number 1, see multiplication as a transformation that changes the number: If they can't think it through, Euler's formula is still a magic spell to them.

While writing, I thought a companion video might help explain the ideas more clearly: It follows the post; watch together, or at your leisure. Euler's formula is the latter: If we examine circular motion using trig, and travel x radians: The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle.

Now let's figure out how the e side of the equation accomplishes it. What is Imaginary Growth?

Combining x- and y- coordinates into a complex number is tricky, but manageable. But what does an imaginary exponent mean? Let's step back a bit.

When I see 34, I think of it like this: Regular growth is simple: Imaginary growth is different: It's like a jet engine that was strapped on sideways -- instead of going forward, we start pushing at 90 degrees.

The neat thing about a constant orthogonal perpendicular push is that it doesn't speed you up or slow you down -- it rotates you!

Taking any number and multiplying by i will not change its magnitude, just the direction it points.So our formula for the golden ratio above (B 2 – B 1 – B 0 = 0) can be expressed as this. 1a 2 – 1b 1 – 1c = 0. The solution to this equation using the quadratic formula . Jul 02, · Finding the Center-Radius Form of a Circle by Completing the Square - Example 3.

Category Education; Conics given the endpoints of a circle find the equation - Duration. From the Pythagorean theorem we have the relationship: a² + b² = c² In our case we will let c = R+δ; b=R; a=d; where R is the radius of the Earth (we will use miles), d is our distance, and δ which is our unknown quantity (the drop height) so we can put these into the equation as follows.

The center-radius form of the circle equation is in the format (x – h) 2 + (y – k) 2 = r 2, with the center being at the point (h, k) and the radius being "r".

This form of the equation is helpful, since you can easily find the center and the radius. Intermediate Algebra Skill Writing the Center-Radius Form of the Equation of a Circle Use the information provided to write the standard form equation of each circle. -Fractal Phase Conjugate Nature of Self-Organization/ LIFE- in TIME: Revealed by Dan Winter's new equation for implosive phase conjugation - precise golden ratio exponents- .

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Differential Equations