Why π^π^π^π could be an integer (for all we know!).

Stand-up Maths
27 Feb 202115:21
EducationalLearning
32 Likes 10 Comments

TLDRThe video script discusses the intriguing mathematical concept of raising pi to the power of pi and whether the result is an integer. It explores the nature of transcendental numbers like pi, the limitations of current computational abilities to calculate such large numbers, and the challenges in proving the result mathematically. The script also highlights the current world records for calculating pi and the potential of Schanuel's Conjecture in providing a theoretical answer, although it remains unproven. The video is sponsored by Jane Street, emphasizing their support for the Stand Up Maths channel and their programs for those interested in mathematical careers.

Takeaways
  • 🤔 The concept of 'pi to the pi' is intriguing and prompts the question of whether the result is an integer, which turns out to be a complex mathematical problem.
  • 📈 Pi is a transcendental number, which means it cannot be written as a simple fraction or the solution to a neat, finite equation.
  • 🔢 Transcendental numbers like pi, e, and the natural log(2) can sometimes result in integers when combined in certain mathematical operations.
  • 🌐 The video mentions Tim Gowers, a renowned mathematician, who also initially thought of calculating 'pi to the pi' but realized it's not feasible.
  • 🧠 Raising an irrational number to the power of another irrational can yield a rational result, as demonstrated by the example of root(2) to the power of root(2).
  • 🔍 Calculating 'pi to the pi' is practically impossible due to the sheer size of the numbers involved and the current limits of computational power.
  • 🏆 The current world record for calculating the most digits of pi stands at 50 trillion digits, which is far from the number of digits needed for 'pi to the pi'.
  • 🛠️ Even with approximations, accurately calculating 'pi to the pi' is challenging because pi has infinitely many digits and we can only know a finite number of them.
  • 📚 Schanuel's Conjecture is mentioned as a potential mathematical tool to address questions about transcendental numbers, but it remains unproven after decades.
  • 🎥 The video was sponsored by Jane Street, a company known for its programs that offer opportunities in solving complex problems in the financial sector.
  • 🎓 The Stand Up Maths channel encourages viewers, especially students and job seekers, to explore the programs offered by Jane Street for career development.
Q & A
  • What was the main topic of the video?

    -The main topic of the video was the mathematical concept of raising pi to the power of pi and whether the result is an integer, and why it's challenging to calculate or prove.

  • Who was the first person to respond to the idea of calculating pi to the pi?

    -Tim Gowers, a Fields Medal mathematician, was the first person to respond to the idea of calculating pi to the pi.

  • What are the three categories of numbers mentioned in the video?

    -The three categories of numbers mentioned are integers, rationals, and irrationals.

  • What is a transcendental number?

    -A transcendental number is a real number that is not the root of any non-zero polynomial equation with rational coefficients, such as pi, e, and the natural log of 2.

  • How did the video demonstrate that irrational numbers can lead to a rational result when raised to powers?

    -The video showed that by raising the square root of 2 to the power of the square root of 2 and then raising that result to the power of the square root of 2 again, you eventually get the number 2, which is rational.

  • What is the current world record for calculating the digits of pi?

    -The current world record for calculating the digits of pi is 50 trillion digits, achieved by Timothy in 2020.

  • Why can't we calculate pi to the pi to the pi to the pi exactly?

    -We can't calculate pi to the pi to the pi to the pi exactly because the number of digits required is in the billions of billions, which is a million times more digits than what we can currently compute, and we only know a finite number of digits of pi.

  • What is Schanuel's Conjecture mentioned in the video?

    -Schanuel's Conjecture is a mathematical proposition that makes statements about the transcendental nature of certain numbers and their combinations, but it is complex and has not been proven.

  • What is the significance of the video's title being 'irrational'?

    -The title being 'irrational' signifies the unexpected and counterintuitive nature of the mathematical concepts discussed, particularly the fact that operations involving irrational numbers can sometimes yield rational results.

  • How does the video relate to potential job opportunities?

    -The video relates to job opportunities by mentioning that Jane Street, the sponsor of the Stand Up Maths channel, offers various programs for university students and those seeking careers in solving complex mathematical problems in the financial world.

  • What was the humorous element in the video regarding the audience's reaction?

    -The humorous element was the audience's reaction to the conclusion that pi to the pi to the pi to the pi is 'just a bit irrational,' which led to them booing, and the speaker jokingly threatening to send them back for a refund.

Outlines
00:00
🤔 The Mystery of Pi to the Power of Pi

The video begins with a discussion on the intriguing mathematical problem of whether pi to the power of pi equals an integer. It introduces the topic with a humorous tone, questioning the validity of such a claim. The video is sponsored by Jane Street, a company with programs for those interested in jobs in mathematics. The speaker acknowledges the immediate thought of calculating the value to verify the claim, a thought shared by renowned mathematician Tim Gowers. However, it's revealed that this calculation is impossible due to the nature of pi being a transcendental number. The speaker then delves into the classification of numbers, distinguishing between integers, rationals, irrationals, and transcendental numbers, setting the stage for a deeper exploration of the problem.

05:05
🧮 Power-Tower Calculations and the Limits of Computation

This paragraph explains the process of calculating power-towers, starting from the top and working downwards. It uses the number 3 as a placeholder for pi to illustrate the concept, showing that the numbers involved become incredibly large very quickly. The speaker contrasts the known world record for calculating pi (31 trillion digits) with the astronomical number of digits required for pi to the power of pi (almost a billion billion digits). The challenges of computation are highlighted, including the need for exact values of pi and the limitations of using approximations. The speaker also discusses the difficulty of the last step in the calculation, where the exact value of pi is crucial, and the current inability to perform such extensive computations.

10:06
🔢 The Truncation of Pi and its Impact on Calculations

The speaker presents a series of experiments and observations on the impact of truncating pi during calculations. It is found that if you raise a truncation of pi to a power that is twice the number of digits in the approximation, no decimal places in the answer will match the true value. This highlights the need for an incredibly precise approximation of pi to obtain an accurate result. The speaker also discusses the limitations of current computational methods and the vast number of decimal places needed to achieve a sufficiently accurate outcome. The paragraph concludes with the admission that we cannot currently calculate or prove whether pi to the power of pi is an integer, pointing to the limitations of our mathematical tools and the potential relevance of Schanuel's Conjecture.

15:10
🎥 Sponsorship and Conclusion of the Mathematical Discussion

The video concludes with a return to the sponsorship by Jane Street, emphasizing their support for the Stand Up Maths channel and their range of programs for those interested in mathematics and the financial world. The speaker encourages viewers to explore these opportunities and appreciates Jane Street's contribution to making the channel possible. The video ends with a mention of the Stand Up Maths production and a canned studio audience, wrapping up the mathematical discussion with a touch of humor.

Mindmap
Keywords
💡Transcendental numbers
Transcendental numbers are a category of real numbers that are not the root of any non-zero polynomial equation with rational coefficients. They cannot be expressed as a ratio of integers or as the solution to a finite algebraic equation. In the video, pi (π) and e are mentioned as examples of transcendental numbers, which are surprising because their certain combinations and powers can result in integers, despite their complex nature.
💡Irrational numbers
Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. They have non-repeating, non-terminating decimal expansions. The square root of 2 (√2) is an example of an irrational number. In the context of the video, it's shown that raising an irrational number to the power of another irrational number can result in a rational number, and even an integer in some cases.
💡Rational numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. They include all integers, as well as fractions like 1/2 or 5/8. Rational numbers can be written as a ratio, which is reflected in the term 'rational', derived from the Latin word 'ratio'. In the video, rational numbers are contrasted with irrational and transcendental numbers to highlight the different behaviors and properties of numbers when raised to high powers.
💡Power-tower
A power-tower is a mathematical expression where an exponent is itself raised to another exponent, creating a nested structure of exponentiation. For example, (3^3)^3 is a power-tower because the exponent 3 is 'raised to the power of' another exponent 3. In the video, the concept of a power-tower is used to discuss the complexity of calculating pi raised to the power of pi, and the resulting number's immense size.
💡Schanuel's Conjecture
Schanuel's Conjecture is an unsolved problem in number theory that proposes conditions under which the exponential of a linear combination of algebraic and transcendental numbers remains transcendental. It is a complex conjecture involving field extensions and other advanced mathematical concepts. The video mentions Schanuel's Conjecture as a potential tool to understand whether pi to the power of pi is transcendental, but acknowledges that it remains unproven after more than half a century.
💡Computational complexity
Computational complexity refers to the amount of resources required to solve a computational problem, such as time and computational power. In the context of the video, it highlights the immense computational effort required to calculate pi to the power of pi due to the incredibly large number of digits involved, which is beyond the current capabilities of even the most powerful computers.
💡Fields medal
The Fields Medal is the most prestigious award in the field of mathematics, often referred to as the 'Nobel Prize of Mathematics'. It is awarded to mathematicians who have made significant contributions to their field. In the video, the mention of a Fields medalist, Tim Gowers, underscores the seriousness and depth of the mathematical discussion regarding the calculation of pi to the power of pi.
💡Jane Street
Jane Street is a global trading firm and the principal sponsor of the Stand Up Maths channel. The company is known for its quantitative and technology-driven approach to finance. In the video, various programs offered by Jane Street for university students and those seeking careers in finance are discussed, emphasizing their role in providing opportunities for individuals interested in solving complex mathematical problems.
💡Approximation
An approximation is a value that is close to but not exactly equal to a given number or expression. In mathematics, approximations are often used when the exact value is unknown or difficult to compute. In the context of the video, the concept of approximation is crucial when discussing the limitations of calculating pi to the power of pi, as only a finite number of decimal places of pi are known, requiring the use of approximations that can affect the accuracy of the result.
💡Mathematical conjecture
A mathematical conjecture is a statement or proposition that is believed to be true but has not yet been proven or disproven. Conjectures often arise from patterns observed in mathematical calculations or from theoretical considerations. In the video, the discussion of Schanuel's Conjecture highlights the existence of unproven ideas in mathematics that could potentially shed light on the properties of transcendental numbers like pi.
Highlights

The video explores the surprising concept of raising pi to the power of pi and whether the result is an integer.

The idea was sparked by a tweet that caught the attention of Tim Gowers, a Fields Medallist mathematician.

Pi is a transcendental number, which makes the possibility of it resulting in an integer when raised to itself surprising.

Rational numbers include integers and can be written as fractions, while irrationals cannot be expressed as fractions but can be solutions to equations.

Transcendental numbers like pi, e, and the natural log of 2 cannot be written as solutions to finite equations and require infinite series for their representation.

The video demonstrates that raising an irrational number to the power of another irrational can result in a rational number, and even an integer.

The video humorously suggests setting pi equal to 3 for simplicity, but emphasizes the importance of working from the top down in power-towers.

Calculating pi to the pi to the pi to the pi is practically impossible due to the sheer number of digits involved.

The current world record for calculating digits of pi is 50 trillion digits, achieved by Timothy in 2020.

The video explains the limitations of using approximations of pi in calculations, as the accuracy decreases significantly with each step.

A rough conjecture suggests that raising a truncation of pi to a power twice the number of digits in the approximation results in no correct decimal places in the answer.

The video highlights the lack of mathematical tools to prove whether pi to the power of pi is transcendental or not.

Schanuel's Conjecture offers a glimmer of hope for proving the transcendental nature of certain combinations of numbers, but it remains unproven after decades.

The video concludes that until Schanuel's Conjecture is proven or computing power significantly advances, the question of whether pi to the pi to the pi to the pi is an integer remains unanswered.

The video was sponsored by Jane Street, a company offering programs for those interested in careers involving complex problem-solving in finance.

Jane Street sponsors the Stand Up Maths channel, making the production of educational mathematical content possible.

The video was filmed in front of a canned studio audience sample pack, adding a unique element to the production.

Transcripts
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