- #1
jgens
Gold Member
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- 50
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!
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.
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.
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.
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.
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.