Download Number: From Ahmes to Cantor by Midhat Gazalé PDF
By Midhat Gazalé
This booklet presents a transparent exposition of the background of numbers. I want I had learn this ebook in the course of my university days. i discovered it valuable in realizing rational, irrational, transcendental numbers, and so on. Mr. Gazale has provided an unique definition of genuine numbers. His generalization of Euler's Theorem is novel and robust, but so uncomplicated. This publication can be a foundation for math classes at faculties and universities national.
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Additional resources for Number: From Ahmes to Cantor
Example text
We begin with a few intuitively appealing properties of convergent sequences which will be needed later. First, a definition. 4 A sequence aj is said to be bounded if there is a number M > 0 such that |aj | ≤ M for every j. 5 Let {aj } be a convergent sequence. Then we have: • The limit of the sequence is unique. • The sequence is bounded. Proof: Suppose that the sequence has two limits α and α. Let ǫ > 0. Then there is an integer N > 0 such that for j > N we have the inequality |aj −α| < ǫ. Likewise, there is an integer N > 0 such that for j > N we have |aj − α| < ǫ.
Continue in this fashion, halving the interval, choosing a half with infinitely many sequence elements, and selecting the next subsequential element from that half. Let us analyze the resulting subsequence. Notice that |aj1 − aj2 | ≤ M since both elements belong to the interval [0, M ]. Likewise, |aj2 − aj3 | ≤ M/2 since both elements belong to [0, M/2]. In general, |ajk − ajk+1 | ≤ 2−k+1 · M for each k ∈ N. Now let ǫ > 0. Choose an integer N > 0 such that 2−N < ǫ/(2M ). Then, for any m > l > N we have |ajl − ajm | = |(ajl − ajl+1 ) + (ajl+1 − ajl+2 ) + · · · + (ajm−1 − ajm )| ≤ |ajl − ajl+1 | + |ajl+1 − ajl+2 | + · · · + |ajm−1 − ajm | ≤ 2−l+1 · M + 2−l · M + · · · + 2−m+2 · M = 2−l+1 + 2−l + 2−l−1 + · · · + 2−m+2 · M = = (2−l+2 − 2−l+1 ) + (2−l+1 − 2−l ) + .
10. Use the Fundamental Theorem of Algebra to prove that any polynomial of degree k has k (not necessarily distinct) roots. 11. Prove that the complex roots of a polynomial with real coefficients occur in complex conjugate pairs. 12. Calculate the square roots of i. 13. In the complex plane, draw a picture of S = {z ∈ C : |z − 1| + |z + 1| = 2} . 14. In the complex plane, draw a picture of T = {z ∈ C : |z + z| − |z − z| = 2} . 15. Prove that any nonzero complex number has kth roots r1 , r2 , . .