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TagHyperreal numbers

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Can We Add New Numbers to Mathematics?

We can work with hyperreal numbers using conventional methods. It could start in high school

Sometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system. In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio. There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios…

An abstract computer generated fractal design. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales.

Are Divergent Series Really an “Invention of the Devil”?

The real villain in the piece is horrendously non-specific concepts of infinity. But that can be fixed

It turns out that hyperreal numbers (i.e., infinities that obey algebraic rules) resolve many of the paradoxes that previously plagued conceptions of divergent series. It is now possible to assign specific values to divergent series.

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Yes, You Can Manipulate Infinity—in Math

The hyperreals are bigger (and smaller) than your average number — and better!

Hyperreal numbers are a new type of number that was developed to simplify and rethink the way that we deal with very large and very small numbers.

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