1: Number Representation in Different Bases and Cryptography
- Page ID
- 181958
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Hindu-Arabic Positional System
- This section covers the Hindu-Arabic numeral system's origin, the significance of exponentiation and place value in modern arithmetic, and the distinction between numerals and numbers. It explains converting numbers to and from expanded form, highlights the role of place value, and provides examples and exercises for practice. Additionally, it introduces historical figures in the numeral system's development and includes questions for assessing understanding of these concepts.
- 1.2: Early Numeration Systems
- This section examines ancient numeral systems: Babylonian, Mayan, and Roman, focusing on their conversion to Hindu-Arabic numerals. It explains the Babylonian sexagesimal system and the absence of zero, along with conversion methods highlighting place values. The Mayan base-20 system uses zero and repeating symbols, while Roman numerals employ specific rules for representation and subtraction, lacking place value.
- 1.3: Converting to Different Base Systems
- This section provides a comprehensive overview of base number systems, covering conversions between base ten and other bases like two, four, five, six, seven, eight, nine, and twelve. It introduces positional base systems, highlights the significance of digit placement, and explains valid digits for each base. The process for conversions involves systematic methods such as identifying powers of the base, using the division method, and collecting remainders.
- 1.4: Addition and Subtraction in Base Systems
- This section focuses on arithmetic in different numeral systems, specifically bases 2 through 9 and 12. It explains how computers use base 2 (binary) for calculations and how conventional base 10 arithmetic changes with non-decimal bases. It details the construction of addition tables for these bases and illustrates addition and subtraction through examples. Key examples include calculations in bases 6, 7, and 12, and it also highlights common errors.
- 1.5: Cryptography
- This section explores various cryptographic techniques, including substitution and transposition ciphers like the Caesar and Bifid ciphers. It emphasizes the significance of key security and the vulnerabilities of these methods to frequency analysis. The evolution of encryption standards from DES to AES is highlighted, showcasing improvements in security.
- 1.6: Modular Arithmetic
- This section explores modular arithmetic, or clock arithmetic, emphasizing its practical applications in scenarios like time calculations and scheduling. It explains the concept of modulus, computation with negative numbers, and includes examples and exercises, particularly focusing on modulo 24 for hours, modulo 7 for days, and modulo 12 for months.