# Construction of the Real Numbers

• jgens
In summary, there are several resources for learning about the construction of the real numbers via Cauchy sequences. Rudin's Principals of Mathematical Analysis has it as an appendix in its first chapter, but using a different approach. Spivak's Calculus includes it as an exercise with guidance, while Terence Tao's lecture notes provide a more in-depth explanation. Another book, Thurston's Topology and Geometry, also covers this topic, and Axiomatic Set Theory by Suppes has a section on it as well.
jgens
Gold Member
Does anyone have or know of any good books that cover the construction of the real numbers via cauchy sequences? I would appreciate any recommendations. Thanks!

rudin doesn't use cauchy sequences - it uses dedekind cuts.

Hmm, I think Spivak includes this construction as an exercise. IIRC, he guides you along so you get the basic idea, but the tricky parts such as demonstrating the existence of a multiplicative inverse is left entirely to you. If you want to just read about it, Terence Tao's lecture notes might be helpful:

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf

Axiomatic Set Theory by Suppes. You can browse through it with Google Books, see page 188.

## What are the real numbers?

The real numbers are a set of numbers that includes all rational and irrational numbers. They can be represented on a number line and are used to measure continuous quantities.

## Why is the construction of real numbers important?

The construction of real numbers is important because it provides a rigorous and logical foundation for understanding and working with numbers. It allows for precise and accurate calculations and is essential in many areas of mathematics and science.

## How are real numbers constructed?

Real numbers are constructed using a process called Dedekind cuts, which involves dividing the rational numbers into two sets based on a specific property. This process ensures that every real number has a unique representation and follows the axioms of the real number system.

## What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a ratio of two integers and have a finite or repeating decimal representation. Irrational numbers, on the other hand, cannot be expressed as a ratio and have an infinite, non-repeating decimal representation. Examples of irrational numbers include pi and the square root of 2.

## How are real numbers used in science?

Real numbers are used in science to represent and measure continuous quantities such as length, time, and temperature. They are also used in mathematical models and equations to describe natural phenomena and make predictions. Real numbers are essential in fields such as physics, chemistry, and engineering.

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