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BustedBreaks
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I was wondering if someone could recommend a good text that explains the construction of complex from real, real from rational, rational from integers, and integers from natural numbers.
Thanks
Thanks
Thanks, I think I'll pick up Thurston's book eventually.Petek said:Here are two books that cover what you asked for:
https://www.amazon.com/dp/082182693X/?tag=pfamazon01-20 by Edmund Landau
https://www.amazon.com/dp/0486458067/?tag=pfamazon01-20 by H. A. Thurston
Landau's book is well-known. However, as some of the Amazon reviews point out, it is extremely dry. It contains little, if any, motivational material and no exercises.
Thurston, on the other hand, consists of approximately one-half motivational material and one-half formal definitions, theorems and proofs. It also contains exercises, but no solutions.
Landau uses Dedekind cuts to define the real numbers, while Thurston uses Cauchy sequences. Both approaches are valid, but the Dedekind cuts approach is probably more commonly seen elsewhere.
So, you might want to see if your library has both books and decide which one you like the most.
Petek
Complex numbers are a type of number that includes both a real and an imaginary component. They are typically written in the form a + bi, where a represents the real part and bi represents the imaginary part. The imaginary unit i is defined as the square root of -1.
The concept of complex numbers was first introduced by Italian mathematician Gerolamo Cardano in the 16th century, but it wasn't until the 18th century that mathematicians began to fully understand and utilize them. The development of complex numbers was heavily influenced by the study of polynomial equations and the desire to find solutions for equations that could not be solved using only real numbers.
Complex numbers have many properties, including the commutative, associative, and distributive properties, similar to real numbers. They also have their own unique properties, such as the conjugate property where the conjugate of a complex number a + bi is a - bi. Additionally, complex numbers have a geometric interpretation as points on the complex plane.
Complex numbers have many applications in mathematics, including in the study of polynomials, differential equations, and signal processing. They are also used in physics, engineering, and other scientific fields to model and solve problems that involve both real and imaginary quantities.
Yes, complex numbers have many practical uses in fields such as electrical engineering, where they are used to represent and analyze alternating currents and circuits. They are also used in computer graphics, quantum mechanics, and other areas of science and technology.