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Two formulas for pi, very different calculation costs

Correct formulas for pi can require very different amounts of calculation. In an arctangent series, small fractional inputs make successive powers shrink quickly, helping explain why a longer-looking formula can need fewer terms.

5–7 min

Questions you could explore

  • What should count as a fair comparison of computational work?
  • How does a series' rate of convergence affect the work needed to calculate another digit of pi?

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  • Wolfram MathWorldMachin’s Formula

    Gives the identity combining arctangents of 1/5 and 1/239. It provides a specific formula for comparing convergence, rather than a claim about today’s fastest pi algorithm.

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Maths and statistics

When one long wait changes the average

An average wait can sound straightforward until one person waits much longer than everyone else. The mean and median offer two ways to summarize that same group. Their different responses to an unusually large value raise a practical question: what do we want an average to tell us?

5–7 min

When one long wait changes the average

Maths and statistics5–7 minAn average wait can sound straightforward until one person waits much longer than everyone else. The mean and median offer two ways to summarize that same group. Their different responses to an unusually large value raise a practical question: what do we want an average to tell us?

Maths and statistics

How is random sampling different from random assignment?

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3–5 min

How is random sampling different from random assignment?

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Maths and statistics

The leap years the Gregorian calendar skips

Most people learn that a leap year happens every four years, yet the year 1900 had no February 29 while the year 2000 did. Because Earth's orbit takes slightly less than 365.25 days, the Gregorian calendar skips three leap years every four centuries to keep the calendar aligned with the seasons.

2–3 min

The leap years the Gregorian calendar skips

Maths and statistics2–3 minMost people learn that a leap year happens every four years, yet the year 1900 had no February 29 while the year 2000 did. Because Earth's orbit takes slightly less than 365.25 days, the Gregorian calendar skips three leap years every four centuries to keep the calendar aligned with the seasons.

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