p-adic Numbers (Alternative Notion of Closeness)
p-adic numbers define a different metric based on divisibility by powers of a prime p, leading to alternative concepts of convergence. For instance, the series 1+2+4+8+... converges to -1 in the 2-adic sense, illustrating how the notion of 'closeness' can be reinterpreted.
THE IDEA
Source Notes and p-adic Closeness
George Berkeley, A Discourse Addressed to an Infidel Mathematician (1734), ed. David R. Wilkins, www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf (accessed Jan. 13, 2014).
Most people, if you press them : David O. Tall and Rolph L. E. Schwarzenberger, “Conflicts in the Learning of Real Numbers and Limits,” Mathematics Teaching 82 (1978): 44–49.
2-adic numbers : In Cauchy’s theory, a series converging to a limit x means that when you sum more and more terms, the total gets closer and closer to x. This requires that we have in mind an idea of what it means for two numbers to be “close” to each other. It turns out that the familiar notion of closeness is not the only one! In the 2-adic world, two numbers are said to be close to each other when their difference is a multiple of a large power of 2. When we say the series 1 + 2 + 4 + 8 + 16 + . . . converges to −1, we are saying that the partial sums 1, 3, 7, 15, 31 . . . are getting closer and closer to −1. With the usual meaning of “close,” that’s not true; but using 2-adic closeness, it’s a different story. The numbers 31 and −1 differ by 32, which is 2^5, a pretty small 2-adic number. Sum a few more terms and you get 511, which differs from –1 by only 512, smaller (2-adically) still. Much of the math you know—calculus, logarithms and exponentials, geometry—has a 2-adic analogue (and indeed a p-adic analogue for any p), and the interaction between all these different notions of closeness is a crazy and glorious story of its own.
including the Italian mathematician/priest Guido Grandi : Material on Grandi and his series is largely drawn from Morris Kline, “Euler and Infinite Series,” Mathematics Magazine 56, no. 5 (Nov. 1983): 307–14.
Cauchy was interested in the truth : The story of Cauchy’s calculus class is taken from Duel at Dawn, Amir Alexander’s immensely interesting historical study of the interaction between math and culture at the beginning of the nineteenth century. See also Michael J. Barany, “Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor,” Notices of the American Mathematical Society 60, no. 10 (Nov. 2013): 1334–38, for a somewhat contrary point of view concerning the modernness of Cauchy’s approach.
When to use it
In number theory, p-adic numbers are used to solve Diophantine equations and study rational numbers.
In cryptography, p-adic integers have applications in some algorithms and constructions.
In mathematical physics, p-adic strings and p-adic quantum mechanics explore alternative spacetime structures.
In analysis, p-adic analogues of calculus, logarithms, and exponentials provide insights into the interplay between different notions of continuity and differentiability.
FROM THEORY TO EVERYDAY LIFE
In practice
Convergence of 1+2+4+8+... to -1 in 2-adic metric
In the standard real metric, the series 1+2+4+8+... diverges to infinity. However, in the 2-adic metric, where distances are measured by powers of 2, the partial sums 1, 3, 7, 15, 31, ... approach -1. For instance, the difference between 31 and -1 is 32 = 2^5, which is considered small in the 2-adic sense, and as more terms are added, this difference becomes an even higher power of 2, making the sum converge to -1.
A MOMENT TO REFLECT
Put it to work.
How does redefining 'closeness' change the behavior of infinite series? Can you think of other contexts (e.g., digital signals, quantum mechanics) where a different notion of distance might be useful?
The p-adic metric is based on divisibility by powers of a prime. How does this property lead to alternative forms of calculus and analysis?
Consider the series 1+2+4+8+... in both the real and 2-adic metrics. What does this example teach you about the dependence of convergence on the underlying metric?
Often confused with
References
- A Discourse Addressed to an Infidel Mathematician — George Berkeley (1734)
- Conflicts in the Learning of Real Numbers and Limits — David O. Tall and Rolph L. E. Schwarzenberger (1978)
- Euler and Infinite Series — Morris Kline (1983)
- Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics — Amir Alexander (2010)
- Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor — Michael J. Barany (2013)
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)