BIS 2A Lecture Notes - Lecture 1: Logarithm, Graph Of A Function, Quadratic Formula

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maroonmammoth784

4 Jul 2018

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Algebra Cheat Sheet

Basic Properties & Facts

Arithmetic Operations

( )

bab

abacabca

aaaac

cbcb

acadbcacadbc

bdbdbdbd

abbaabab

cddcccc

abacad

bca c

abc



+=+=

















+−

+=−=

−−+

==+

−−





+

=+≠=







Exponent Properties

( )

1, 0

nmnmnm

mmn

nnm

nnn

aaaa

abab bb

abb

aaa

baa

+−

−

===

==≠









====





Properties of Radicals

,if is odd

,if is even

nnnn

nnm nn

n n

aaabab

aa b

aan

Properties of Inequalities

If thenand

If and 0 then and

abacbcacbc

abcacbc

<+<+−<−

<><<

<<>>

Properties of Absolute Value

if 0

≥





−<



Triangle Inequality

aaa

abab bb

abab

≥−=

+≤+

Distance Formula

(

)

111

Pxy

= and

(

)

222

Pxy

= are two

points the distance between them is

( ) ( ) ( )

122121

dPPxxyy

=−+−

Complex Numbers

( ) ( ) ( )

()( ) ( )

( )( )

( )

( )( )

11,0

Complex Modulus

Complex Conjugate

iiaiaa

abicdiacbdi

abicdiacbdadbci

abiabiab

abiab

abiabi

abiabiabi

=−=−−=≥

+++=+++

+−+=−+−

++=−++

+−=+

+=+

+=−

++=+

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Document Summary

Exponent properties n m a a a n m n a m a n m a. 0 n n n a b n a b n a n b n a. 1 a n n n n n b b a a b. 1 n a n ab n n a b m n a nm a n a b n n a b n n n a n a a. , if n is even m n a ab. Properties of absolute value a c a c b c b c if a if a. 0 a a a ab a b a a a b a b a b a b. 2 are two points the distance between them is d p p. 1 i a a d i d i ac bd ad bc i a c b. 2 i c di c di c di b.

Related Questions

1.Write a function encipher(s, n) that takes as inputs an arbitrary string s and a non-negative integer n between 0 and 25, and that returns a new string in which the letters in s have been "rotated" by n characters forward in the alphabet, wrapping around as needed. For example:

>>> encipher('hello', 1) result: 'ifmmp'

>>> encipher('hello', 2) result: 'jgnnq'

>>> encipher('hello', 4) result: 'lipps'

Upper-case letters should be "rotated" to upper-case letters, even if you need to wrap around. For example:

>>> encipher('XYZ', 3)

result: 'ABC'

Lower-case letters should be "rotated" to lower-case letters:

>>> encipher('xyz', 3)

result: 'abc'

Non-alphabetic characters should be left unchanged:

>>> encipher('#caesar!', 2)

result: '#ecguct!'

Hints/reminders:

You can use the built-in functions ord and chr convert from single-character strings to integers and back:

>>> ord('a')

result: 97

>>> chr(97)

result: 'a'

You can use the following test to determine if a character is between 'a' and 'z' in the alphabet:

if 'a' <= c <= 'z':

A similar test will work for upper-case letters.

We recommend writing a helper function rot(c, n) that rotates a single character c forward by n spots in the alphabet. We have given you a template for this helper function in ps3pr3.py that checks to ensure that c is a single-character string. We wrote rot13(c) in lecture; rot(c, n) will be very close to rot13(c)! You can test your rot(c, n) as follows:

>>> rot('a', 1)

result: 'b'

>>> rot('y', 2)

result: 'a'

>>> rot('A', 3)

result: 'D'

>>> rot('Y', 3)

result: 'B'

>>> rot('!', 4)

result: '!'

Once you have rot(c, n), you can write a recursive encipher function.

Once you think you have everything working, here are three more examples to try:

>>> encipher('xyza', 1)

result: 'yzab'

>>> encipher('Z A', 2)

result: 'B C'

>>> encipher('Caesar cipher? I prefer Caesar salad.', 25)

result: 'Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.'

2.Write a function decipher(s) that takes as input an arbitrary string s that has already been enciphered by having its characters "rotated" by some amount (possibly 0). decipher should return, to the best of its ability, the original English string, which will be some rotation (possibly 0) of the input string s. For example:

>>> decipher('Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.')

result: 'Caesar cipher? I prefer Caesar salad.'

Here are two more examples:

>>> decipher('Hu lkbjhapvu pz doha ylthpuz hmaly dl mvynla lclyfaopun dl ohcl slhyulk.')

result: 'An education is what remains after we forget everything we have learned.'

>>> decipher('python')

result: 'eniwdc'

------------------------------------

def rot(c, n):
""" your docstring goes here """
# check to ensure that c is a single character
assert(type(c) == str and len(c) == 1)

def letter_prob(c):
""" if c is the space character (' ') or an alphabetic character,
returns c's monogram probability (for English);
returns 1.0 for any other character.
adapted from:

"""
# check to ensure that c is a single character
assert(type(c) == str and len(c) == 1)

if c == ' ': return 0.1904
if c == 'e' or c == 'E': return 0.1017
if c == 't' or c == 'T': return 0.0737
if c == 'a' or c == 'A': return 0.0661
if c == 'o' or c == 'O': return 0.0610
if c == 'i' or c == 'I': return 0.0562
if c == 'n' or c == 'N': return 0.0557
if c == 'h' or c == 'H': return 0.0542
if c == 's' or c == 'S': return 0.0508
if c == 'r' or c == 'R': return 0.0458
if c == 'd' or c == 'D': return 0.0369
if c == 'l' or c == 'L': return 0.0325
if c == 'u' or c == 'U': return 0.0228
if c == 'm' or c == 'M': return 0.0205
if c == 'c' or c == 'C': return 0.0192
if c == 'w' or c == 'W': return 0.0190
if c == 'f' or c == 'F': return 0.0175
if c == 'y' or c == 'Y': return 0.0165
if c == 'g' or c == 'G': return 0.0161
if c == 'p' or c == 'P': return 0.0131
if c == 'b' or c == 'B': return 0.0115
if c == 'v' or c == 'V': return 0.0088
if c == 'k' or c == 'K': return 0.0066
if c == 'x' or c == 'X': return 0.0014
if c == 'j' or c == 'J': return 0.0008
if c == 'q' or c == 'Q': return 0.0008
if c == 'z' or c == 'Z': return 0.0005
return 1.0

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BIS 2A Lecture Notes - Lecture 1: Logarithm, Graph Of A Function, Quadratic Formula

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