Algebra: An Algorithmic Treatment
by Kenneth E. Iverson
published by Addison-Wesley, 1972 (preliminary edition entitled Elementary Algebra published by IBM, 1971)
Contents
Chapter 1. The Language of Mathematics
1.1 Introduction
1.2 Expressions and Results
1.3 Names
1.4 Over Notation
1.5 The Positive Integers
1.6 Vectors
1.7 Repetitions
1.8 Summary
Chapter 2. Function Tables and Maps
2.1 Introduction
2.2 Reading Function Tables
2.3 Expressions for Producing Function Tables
2.4 The Functions Denoted by ⌈ and ⌊
2.5 The Power Function
2.6 Maps
Chapter 3. The Negative Numbers
3.1 Subtraction
3.2 Negative Integers
3.3 Addition and Subtraction
3.4 Expressions for the Integers
Chapter 4. Function Tables with Negative Integers
4.1 Introduction
4.2 Subtraction
4.3 Flipping Tables
4.4 Indexing Tables
4.5 Addition
4.6 Multiplication
4.7 Maximum and Minimum
4.8 Relations
4.9 Logical Values
4.10 The Over Function on Tables
Chapter 5. The Rational Numbers
5.1 Introduction
5.2 Long Division
5.3 Rational Numbers
5.4 Addition of Rationals Having the Same Divisor
5.5 Multiplication of Rational Numbers
5.6 Multiplication of a Rational by an Integer
5.7 Multiplication Expressed in Terms of Vectors
5.8 Addition of Rationals
5.9 Addition of Rationals in Terms of Vectors
5.10 The Quotient of Two Rationals
5.11 Decimal Fractions
5.12 Addition and Subtraction of Decimal Fractions
5.13 The Decimal Fraction Representation of a Rational
5.14 Decimal Fraction Approximations to Rationals
5.15 Multiplication of Decimal Fractions
5.16 Division of Decimal Fractions
5.17 Exponential Notation
5.18 Division with Negative Arguments
5.19 Division by Zero
Chapter 6. Function Tables with Rational Numbers
6.1 Introduction
6.2 Catenation
6.3 Division Tables
6.4 Comparison
6.5 The Power Function for Negative and Zero Arguments
6.6 The Power Function for Rational Arguments
Chapter 7. The Residue Function and Factoring
7.1 The Residue Function
7.2 Negative Right Arguments
7.3 Divisibility
7.4 Factors
7.5 Compression
7.6 Prime Numbers
Chapter 8. Monadic Functions
8.1 Introduction
8.2 Negation
8.3 Reciprocal
8.4 Magnitude
8.5 Floor and Ceiling
8.6 Complement
8.7 Ravel
8.8 Size
Chapter 9. Function Definition
9.1 Introduction
9.2 Definition of Dyadic Functions
9.3 A Function to Generate Primes
9.4 Temperature Scale Conversion Function
9.5 Functions on Rationals
9.6 Tracing Function Execution
Chapter 10. The Analysis of Functions
10.1 Introduction
10.2 Maps
10.3 Graphs
10.4 Interpreting a Linear Graph
10.5 The Take and Drop Functions
10.6 Difference Tables
10.7 Fitting Functions of the Form a+b×x
10.8 Factorial Polynomials
10.9 Multiplication and Addition of Difference Tables
10.10 Difference Tables for the Factorial Polynomials
10.11 Expressions for Graphs
10.12 Character Vectors
Chapter 11. Inverse Functions
11.1 Introduction
11.2 Inverse of the Function a+b×x
11.3 Difference Tables
11.4 Maps
11.5 Graphs
11.6 Determining the Inverse for a Specific Argument
11.7 The Solution of Equations
Chapter 12. Iterative Processes
12.1 Introduction
12.2 General Root Finder
12.3 Greatest Common Divisor
12.4 The Binomial Coefficients
Chapter 13. Inner Product and Polynomials
13.1 Introduction
13.2 The Inner Product of Two Vectors
13.3 Matrices
13.4 Inner Product with Matrix Arguments
13.5 Polynomials
13.6 Polynomials Expressed as Inner Products
Chapter 14. Identities
14.1 Introduction
14.2 Commutativity
14.3 Associativity
14.4 Distributivity
14.5 Identities Based on Commutativity, Associativity, and Distributivity
14.6 Identities on Vectors
14.7 The Power Function
Chapter 15. Identities on Polynomials
15.1 Introduction
15.2 The Sum of Polynomials
15.3 The Product of Polynomials
15.4 The Product ×/x+v
15.5 Binomial Coefficients
15.6 The Factorial Polynomials
15.7 Mathematical Induction
Chapter 16. The Representation of Numbers
16.1 Introduction
16.2 The Prime Factors System
16.3 The Decimal System
16.4 The Binary System
16.5 Positional Number Systems
16.6 Addition and Multiplication Tables
16.7 Negative Integers
16.8 Rational Numbers
Chapter 17. Logic and Sets
17.1 Logic
17.2 Sets
Chapter 18. Linear Functions
18.1 Introduction
18.2 Mappings
18.3 Rotations
18.4 Translation
18.5 Linear Function on a Set of Points
18.6 Rotation and Translation
18.7 Stretching
18.8 Identities on the Inner Product +.×
18.9 Linear Functions on 3-Element Vectors
18.10 Rotations in Three Dimensions
Chapter 19. Inverse Linear Functions
19.1 Introduction
19.2 Some Inverse Functions
19.3 The Solution of Linear Equations
19.4 Basic Solutions
19.5 Determining Basic Solutions
19.6 Simplified Calculations for Basic Solutions
19.7 The Determinant Function
19.8 Matrix Form of the Basic Solutions
19.9 The General Solution from the Basic Solutions
19.10 The Inverse Linear Functions
19.11 Properties of the Inverse Linear Function
19.12 Alternative Derivation of the Inverse
19.13 Efficient Solution of a Linear Equation
19.14 Inverse Linear Functions in Three Dimensions
19.15 The Inverse Function
19.16 Curve Fitting
Exercises
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Appendix A. Algebra as a Language
A.1 Introduction
A.2 Arithmetic Notation
A.3 Algebraic Notation
A.4 Analogies with the Teaching of Natural Language
A.5 A Program for Elementary Algebra
Appendix B. The Mechanics of Computer Use
B.1 Introduction
B.2 Getting Started
B.3 Using APL
B.4 Error Reports
B.5 Revising a Function Definition
B.6 The Active Workspace
B.7 Terminating a Work Session
B.8 Use of Libraries
Appendix C. User of the Computer in Teaching
C.1 Introduction
C.2 Experimentation
C.3 Checking Solutions to Exercises
C.4 Use as a Computational Tool
C.5 Drill
C.6 Collective Use
C.7 Remote Use
Summary of Notation
Section 1.1 The Language of Mathematics -- Introduction
Algebra is the language of mathematics. It is therefore an essential topic for anyone who wishes to continue the study of mathematics. Moreover, enough of the language of algebra has crept into the English language to make a knowledge of some algebra useful to most non-mathematicians as well. This is particularly true for people who do advanced work in any trade or discipline, such as insurance, engineering, accounting, or electrical wiring. For example, instructions for laying out a playing field might include the sentence, "To verify that the corners are square, note that the length of the diagonal must be equal to the square root of the sum of the squares of the length and the width of the field", or alternatively, "The length of the diagonal must be 
. In either case (whether expressed in algebraic symbols or in the corresponding English words), the comprehension of such a sentence depends on a knowledge of some algebra.
Because algebra is a language, it has many analogies with English. These analogies can be helpful in learning algebra, and they will be noted and explained as they occur. For instance, the integers or counting numbers (1, 2, 3, 4, 5, 6, ...) in algebra correspond to the concrete nouns in English, since they are the basic things we discuss, and perform operations upon. Furthermore, functions in algebra (such as + (plus), - (subtract), and × (times)) correspond to the verbs in English, since they do something to the nouns. Thus, 2+3 means "add 2 to 3" and (2+3)×4 means "add 2 to 3 and then multiply by 4". In fact, the word "function" (as defined, for example, in the American Heritage Dictionary), is descended from an older word meaning, "to execute", or "to perform".
When the language of algebra is compared to the language of English, it is in certain respects much simpler, and in other respects more difficult. Algebra is simpler in that the basic algebraic sentence is an instruction to do something, and algebraic sentences (usually called expressions) therefore correspond to imperative English sentences (such as "Close the door."). For example, 2+3 means "add 2 and 3", and year←1970 means "assign to the name year the value 1970", and y←1970 means "assign to the name y the value 1970". Since imperative sentences form only a small and relatively simple part of English, the language of algebra is in this respect much simpler.
Algebra is also simpler in that it permits less freedom in the ways you can express a particular function. For example, "subtract 2 from 4" would normally be written in algebra only as 4-2 , whereas in English it could be expressed in many ways such as "take the number 2 and subtract it from the number 4", or "compute the difference of the integers 4 and 2".
The most difficult aspect of traditional presentations of algebra is the early emphasis on identities, or the equivalence of different expressions. For example, the expressions (5+7)×(5+7) and (5×5)+(2×5×7)+(7×7) are equivalent in the sense that, although they involve a different sequence of functions, they each yield the same result. English also offers equivalent expressions. For example, "The dog bit the man" is equivalent to "The man was bitten by the dog". It is not that the rules for determining equivalence in algebra are more difficult than in English; on the contrary, they are so much simpler that their study is more rewarding and therefore more attention is given to equivalences in algebra than in English.
In the present treatment this aspect of algebra (that is, the study of identities or equivalence of expressions) is delayed until the student has devoted more attention to the reading, writing, and evaluation of algebraic expressions.
This view of algebra as a language is central to the present treatment. It is buttressed and expanded in Appendix A, and this appendix should perhaps be read first by any teacher and by any student who has significant prior experience with traditional treatments of algebra.
The exercises form an important part of the development, and the point at which the reader should be prepared to attempt each group of exercises is indicated in the margin. For example, the first such marginal note appears as ⌹1-6 and indicates that Exercises 1 to 6 of this chapter may be attempted at that point.
Collections of expressions occurring in certain exercises are broken into groups to provide convenient reference in assigning and discussing exercises. These groups sometimes indicate substantive groupings of the material treated as well.
The exposition and the exercises are organized to encourage experimentation and observation as an essential part of learning. Experimentation and discovery can be further encouraged to a startling degree by the use of an APL computer terminal if one is available. All expressions occurring in the text and exercises can be entered directly on the terminal keyboard without further knowledge of computers. Techniques for the use of the computer in teaching are discussed in Appendix C. Appendix B presents the computer keyboard and other details necessary to putting it in operation.
A student using an APL computer in exploration is sometimes confronted with matters not treated until a later point in the text. For example, a beginning student entering the expression
2000×3000×4000
will receive the response
2.4e10
This result is expressed in exponential notation (meaning 2.4 times 10 to the power 10) which is not discussed until Section 5.17. The Index, the Summary of Notation (appearing inside the covers), and Appendix B can be used to resolve such difficulties.