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Sometimes math students get confused by the notion of whether math is something that someone "made up," or is something that exists in nature, and that mathematicians through history have been "discovering." One example of this would be imaginary numbers. One could argue that in order for the fundamental theorem of algebra to be true, someone decided to invent imaginary numbers. Or one could argue that the first use of imaginary numbers was a great discovery of a type of numbers that already existed, but hadn't been identified yet. How do you look at mathematics? Is it made up or discovered? Explain your answer in a good paragraph..

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Neither made up nor discovered. Mathematics is a language that describes relationships and it changes as discoveries are made. When a new discovery is made, new mathematic operations and variables are added to the mathematic repertoire to describe the relationships found in the new discovery.

Example: Prior to the invention of radio, nobody cared whether or not there was a relationship between wavelength and frequency, and when the principles of electromagnetic radiation were undergoing the process of determining its properties, it was thought that wavelength was more understandable. When amplification principles were employed, wavelength was no longer adequate to compute what sort of circuit design was needed and so the relationship between wavelength and frequency became important mathematically...and so that relationship was established in an equation.

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Excellent!
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In one sense all mathematics is "made up". Hold up one finger, then hold up two. Is this addition? No, but addition can be defined as one more finger makes the next natural number. Imaginary numbers are totally constructed but find application in the world as in quantum mechanics and electrical engineering problems. We use a certain kind of "manifold" to describe physics because it corresponds to the real world where many other manifolds do not. They are not wrong, they just don't describe OUR world. In another sense some of mathematics is discovered, as fractals were. Fractals existed in clouds, shorelines, and leaf outlines and blood vessel branching long before man.

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Umm, fractals are a mathematical feedback loop and so it led to the discovery of natural feedback loops that weren't recognized as such before. This is another feature of mathematics--first it establishes an expression of an observed relationship and then when the stated relationship is manipulated inside an equation to determine properties, properties that weren't thought of as part of the original relationship leads to more discoveries in nature. This is exactly how the discovery of the full extent of our solar system occurred.
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Whoa!
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