MandysNotes

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Lessons

 

 

Below is a table of various distances in various units that are important in astronomy and astrophysics. 

AstroDistanceChart800

Sunday, 09 March 2014 16:59

The Roots of a Polynomial

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Definition:

The roots of a polynomial \(P\) are the solutions of the equation:

\[ P(x) = 0. \]

Sunday, 09 March 2014 16:51

Polynomials of Degree Two

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The product of two linear factors

\[ (x - \alpha)(x - \beta)\]

is a polynomial of degree two:

\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]

Definition:


Polynomials of degree two are also called quadratic polynomials.

 

Definition:


A  polynomial function of a real variable \(x\), is a function defined by the sum:
\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]

Saturday, 08 March 2014 01:36

Linear Functions and Linear Equations

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 A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:

  • for any real number, \(a\), \(f(ax) = a(f(x));\)
  • for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)
Saturday, 08 March 2014 01:31

Binomial Coefficients and Subsets

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For a finite set  \(S\) with \(n\) elements, the total number of subsets of \(S\) with \(k\) elements

\[ ( 0 \leq k \leq n) \]

is

\[ \binom{n}{k}.\]

Friday, 07 March 2014 21:11

The Factorial

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We define \(n\) factorial, written \(n!\), by induction.

For  \(n = 1,\)
\[ n! = 1! = 1. \]
For \((n+1)\),
\[ (n+1)! = (n+1)(n!). \]

Friday, 07 March 2014 21:08

Mathematical Induction

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For a function \(f(n)\) of a natural number \(n,\) we can prove that \(f(n)\) has the property P, for all \(n\), if we can do two things:

  •  prove that P is true for \(f(n)\) when \(n\) = 1;
  • second: prove that if P is true for \(f(n)\), then P is also true for \(f(n+1)\).


This process is called mathematical induction. 

Friday, 07 March 2014 21:05

Binomial Coefficients

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Definition:

For any natural number \(n\), and any natural number \(i\), such that, \( n \geq i \geq 0\):

The binomial coefficient,

\[ \binom{n}{i},  \]

of \(n\) and \(i\) is,

\[ \binom {n}{i} := \frac{n!}{(n-i)!(i)!}. \]

You can download Latex here:

 

http://latex-project.org/ftp.html 

 

This site has valuable information:

 

Tex-Latex Stack Exchange:

 

http://tex.stackexchange.com/

 

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