How real are the real numbers, really?

TL;DR

Achilles running through fractions and irrationals demonstrates the need for real numbers to support reality.

Transcript

Say Achilles wants to run a race. First he’d have to run 1/2 way, then half the rest of the way then, then half of that and so on... but no, this video isn’t about zeno’s paradox. I just wanted to point out, to get to the end, Achilles had to run through every fraction of the length at some point. So here’s the question for this video. Does Achille... Read More

Key Insights

• #️⃣ Achilles running through an infinite number of fractions demonstrates that there is an infinite number of numbers between any two fractions.
• #️⃣ The existence of irrational numbers further expands the infinity of numbers between fractions.
• 😥 The concept of real numbers, with their property of convergence, is necessary to determine values at specific points in mathematics and physics.
• 😒 Mathematicians have explored various methods, including the use of infinitesimals, to make calculus more rigorous.
• #️⃣ The introduction of hyperreals, a set of numbers that fill the gaps between real numbers, offers another approach to formulating calculus.
• 🌍 Real numbers are essential for real-world applications in physics and other fields.
• 🪘 The development of real numbers was a long process in mathematics to ensure the accuracy and rigour of mathematical concepts.

Questions & Answers

Q: How does Achilles' race demonstrate the existence of an infinite number of fractions?

Achilles running through fractions without jumping shows that there are an infinite number of fractions between any two fractions.

Q: What are irrational numbers and how do they relate to fractions?

Irrational numbers cannot be written as fractions but can be represented by an infinite string of numbers. They exist between fractions, adding to the infinite number of numbers.

Q: How did mathematicians solve the problem of determining Achilles' speed at a particular point?

Mathematicians introduced the concept of real numbers, which have the property of convergence. By averaging infinitesimally small time intervals, the speed at a particular point can be determined.

Q: How did mathematicians reformulate calculus using infinitesimals?

In the 60s, a mathematician discovered that infinitesimals can be used to completely reformulate calculus. The method involves adding another infinite set of numbers, called the hyperreals, which fill the gaps between the real numbers.

Summary & Key Takeaways

• Achilles running through fractions without jumping demonstrates that there are an infinite number of fractions between any two fractions.

• Consideration of irrational numbers reveals that there are even more numbers between fractions, leading to the concept of real numbers.

• Real numbers are necessary for real world applications and it took a long process to develop them in mathematics.