# Tag Archives: ℝ

## Measure: Sizing up the Continuum

For those not in the know, here’s what mathematicians mean by the word “measurable”: The problem of measure is to assign a size ≥ 0 to a subset of ℝ. In other words, to answer the question: How big is that? Like, how big … Continue reading

## Nonterminating decimals do not make sense.

The Banach-Tarski paradox proves how f#cked up the real numbers are. Logical peculiarities confuse our intuitions about “length”, “density”, “volume”, etc. within the continuum (ℝ) of nonterminating decimals. Which is why Measure Theory is a graduate-level mathematics course. These peculiarities … Continue reading

## Do numbers exist?

When I was a math teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story. You explain to me what 4 is by pointing to four rocks on the ground, … Continue reading

## Math Infinity

Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop. That’s all. (\$i++ for programmers) Which is why ∞ seems very small to the mind of a mathematician. With projective geometry you can map ℝ to … Continue reading

Stereographic projection is a way of mapping ℝ — which is infinitely long — onto a circle, which has a finite length (circumference). In a sense, the circle is larger than ℝ. That is, you can map injectively every point … Continue reading

## sqrt10

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## Is Calculus Bull*hit?

The hallmark of a calculus course is epsilon-delta proofs. As one moves closer and closer to a point of interest (reducing δ, the distance from the point-of-interest), the phenomenon’s measure is bounded by something times ε, a linear error term. … Continue reading