4: Number Representation and Calculation
- Page ID
- 129480
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.0: Introduction
- The text discusses the evolution of numbering systems, highlighting the widespread use of the Hindu-Arabic system today, which wasn't developed until the 6th or 7th century C.E. Historically, different cultures devised diverse methods for counting and recording quantities, driven by agricultural, religious, and trade needs. Some cultures had simple counting terms, while others had complex systems like the Oksapmin's 27-word system.
- 4.1: Hindu-Arabic Positional System
- This page covers the concept of the Hindu-Arabic numeral system and the role of place value in representing numbers. It provides learning objectives such as evaluating exponential expressions, converting Hindu-Arabic numerals to expanded form, and vice versa. The text explains the historical development of numerals, highlights the importance of place value, and provides examples of converting numbers between different forms.
- 4.2: Early Numeration Systems
- The page discusses different numerical systems developed by ancient cultures, focusing on Babylonian, Mayan, and Roman numerals. It outlines how each system operates and the conversion methods to the modern Hindu-Arabic numeral system. Babylonians used a base-60 system without a zero, Mayans used a base-20 system with a symbol for zero, while the Romans employed additive and subtractive rules without place values.
- 4.3: Converting with Base Systems
- This page provides an overview of number base systems, emphasizing conversion techniques between bases. It outlines the Hindu-Arabic positional system's base 10 and compares it to other historical bases like the Mayans' base 20 and Babylonians' base 60. The page details how to convert numbers from other bases, such as base 6, base 7, base 14, etc., to base 10, and vice versa. Additionally, it discusses potential errors in base conversions, like using "illegal" symbols.
- 4.4: Addition and Subtraction in Base Systems
- The document 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.
- 4.5: Multiplication and Division in Base Systems
- This page discusses arithmetic operations, specifically multiplication and division, in numerical bases other than base 10. It covers the process of creating multiplication tables for various bases like base 6, 7, and 12, using repeated addition according to the specific base rules. It also tackles division in non-decimal bases and identifies common errors such as using incorrect symbols or applying base 10 rules to other bases.
- 4.6: Chapter Summary
- This page outlines the structure of section 4.6, which includes key terms, key concepts, videos, projects, a chapter review, and a chapter test. The content is organized to facilitate learning and assessment within the chapter. There are also settings and dialog features related to classification and learning path adjustments, indicating options to remove learning paths within the guide.