The First Steps in Algebra

George Albert Wentworth, 1894. Ginn & Company, 1894

Provenance and licence

Text: Project Gutenberg eBook #36670, released July 9, 2011. The transcription's licence, as it states it:

This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net

This site quotes excerpts and problems from that transcription and hosts no scans; scans are linked to their archive. The records below (problems, checks, errata) are from pd-books at b3c614f, dedicated to the public domain (CC0), export generated 2026-10-10T11:13:12Z.

Chapters

Introduction 8 excerpts, 8 equations, 165 problems
Simple Equations 6 excerpts, 5 equations, 89 problems
Positive and Negative Numbers 6 excerpts, 11 equations, 82 problems
Addition and Subtraction 7 excerpts, 8 equations, 54 problems
Multiplication and Division 6 excerpts, 4 equations, 131 problems
Multiplication and Division 6 excerpts, 4 equations, 121 problems
Factors 6 excerpts, 9 equations, 224 problems
Common Factors and Multiples 5 excerpts, 0 equations, 56 problems
Fractions 6 excerpts, 5 equations, 114 problems
Fractional Equations 7 excerpts, 7 equations, 112 problems
Simultaneous Equations of the First Degree 8 excerpts, 8 equations, 55 problems
Quadratic Equations 6 excerpts, 4 equations, 65 problems
Arithmetical Progression 7 excerpts, 8 equations, 28 problems
Geometrical Progression 6 excerpts, 5 equations, 16 problems
Square and Cube Roots 7 excerpts, 2 equations, 58 problems

Index of terms 156 terms, by chapter

Related books

The books the concept graph relates to this one, most shared terms first.

A First Book in Algebra 1895
101 shared terms
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A Course of Pure Mathematics 1921
29 shared terms
absolute value, arithmetical mean, denominator, divisibility, equation, factor, geometrical mean, geometrical progression, highest common factor, imaginary root, integer, linear equation, lowest terms, polynomial, power, product, quadratic equation, real root, reciprocal, relatively prime, root, surd, term, variable, addition, division, multiplication, substitution, subtraction
An Introduction to Mathematics 1911
27 shared terms
algebra, arithmetical progression, digit, equality, equation, geometrical progression, imaginary root, implied multiplication, inequality, integer, linear equation, negative number, number, number line, parenthesis, perfect square, positive number, quadratic equation, root, unknown, variable, addition, multiplication, order of operations, quantity, time, unit
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27 shared terms
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Elementary Illustrations of the Differential and Integral Calculus 1899
22 shared terms
algebraic expression, arithmetical progression, arrangement, coefficient, degree, equality, equation, exponent, geometrical progression, negative number, number, power, product, reciprocal, root, sign, variable, solving an equation, substitution, verification, time, unit
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16 shared terms
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14 shared terms
algebra, arithmetical progression, common difference, cube, digit, equation, factor, geometrical progression, imaginary root, quadratic equation, remainder, writing by inspection, time, difference of two squares
Vector Analysis and Quaternions 1906
9 shared terms
algebra, exponent, partial product, product, quotient, reciprocal, square, trinomial, multiplication
Plane Geometry 1899
7 shared terms
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Elements of Plane Trigonometry 1863
3 shared terms
arithmetical mean, geometrical mean, variable
Solid Geometry with Problems and Applications 1919
2 shared terms
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Treatise on Thermodynamics 1903
2 shared terms
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An Elementary Treatise on Electricity 1888
1 shared term
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Spherical Trigonometry, for the Use of Colleges and Schools 1886
1 shared term
verification

Errata (27)

T001:
Printed:
Corrected: #1
In the source: rrections. \newcommand{\DPtypo}[2]{#2} \newcommand{\Add}[1]{
content
Marked by Project Gutenberg's proofreaders in the transcription.
T002:
Printed: inclosed
Corrected: enclosed
In the source: algebraic number which is to be added or subtracted is often
content
Marked by Project Gutenberg's proofreaders in the transcription.
T003:
Printed: :
Corrected: ,
In the source: ook uses commas elsewhere] \Dictum{To Add Algebraic Numbers}
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T004:
Printed: ,
Corrected: .
In the source: ivide $a^{5}$~by~$a^{2}$, $a^{6}$~by~$a^{4}$, $a^{4}$~by~$a$
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T005:
Printed: ,
Corrected: ;
In the source: }{a^{4}} &= \frac{aaaaaa}{aaaa} &&= aa &&= a^{2} &&= a^{6-4}
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T006:
Printed: ,
Corrected: .
In the source: ac{a^{4}}{a} &= \frac{aaaa}{a} &&= aaa &&= a^{3} &&= a^{4-1}
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T007:
Printed: ,
Corrected:
In the source: cient of~$ax^{4}$ will be $-5 + 2$, or~$-3$; the coefficient
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T008:
Printed: .
Corrected:
In the source: + b + c) p \\ &= am + bm + cm + an + bn + cn + ap + bp + cp
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T009:
Printed:
Corrected: \therefore
In the source: . \begin{align*} &+6 - 5 = 1,\quad 6y × (-5y) = -30y^{2}. \\
content
Marked by Project Gutenberg's proofreaders in the transcription.
T010:
Printed:
Corrected: Resolve into factors:
In the source: {2} = (a + 3b - 5c)(a - 3b + 5c)$. \end{Soln} \Exercise{34.}
content
Marked by Project Gutenberg's proofreaders in the transcription.
T011:
Printed: ,
Corrected: .
In the source: um of the first $16$~terms of the series $5$, $7$, $9$,~$11$
spacing-or-punctuation
Marked by Project Gutenberg's proofreaders in the transcription.
T012:
Printed: the
Corrected: then
In the source: y $3(x + y)^{2}$; and if the third term of the root be~$+z$,
content
Marked by Project Gutenberg's proofreaders in the transcription.
G001: Exercise 17, problem 30 (Divide), p. 45
Printed: $-19mg^{2}t^{3}$~by~$57m2^gt^{4}$
Corrected: $-19mg^{2}t^{3}$~by~$57m^{2}gt^{4}$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 45 (leaf n56), reads "30. -19mg^2t^3 by 57m^2gt^4.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G002: Answers, Exercise 19, answer 3, p. 169
Printed: $2a + 3y -8z$
Corrected: $2x + 3y - 8z$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 169 (leaf n180), reads "3. 2x + 3y - 8z.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G003: Answers, Exercise 20, answer 13, p. 169
Printed: $6z - 2y-z$
Corrected: $6x - 2y - z$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 169 (leaf n180), reads "13. 6x - 2y - z.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G004: Answers, Exercise 22, answer 18, p. 170
Printed: $x^{5} - 3x^{4} - x^{3} + 16x^{2} - 21$
Corrected: $x^{5} - 3x^{4} - 3x^{3} + 16x^{2} - 21$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 170 (leaf n181), reads "18. x^5 - 3x^4 - 3x^3 + 16x^2 - 21.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G005: Exercise 29, problem 3, p. 69
Printed: \dfrac{1 - 27c^{3}}{1 - 2c}
Corrected: \dfrac{1 - 27c^{3}}{1 - 3c}
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 69 (leaf n80), reads "3. (1 - 27c^3)/(1 - 3c).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G006: Answers, Exercise 29, answer 17, p. 172
Printed: $4a^{2}b^{2}c^{2} - 6abc + 9$
Corrected: $4a^{2}b^{2}c^{2} + 6abc + 9$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 172 (leaf n183), reads "17. 4a^2b^2c^2 + 6abc + 9.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G007: Answers, Exercise 34, answer 20, p. 174
Printed: $(5c + 3a - 2x)(4c - 3a + 2x)$
Corrected: $(5c + 3a - 2x)(5c - 3a + 2x)$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 174 (leaf n185), reads "20. (5c + 3a - 2x)(5c - 3a + 2x).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G008: Answers, Exercise 35, answer 16, p. 174
Printed: $(4x^{2} - 3y^{5})(16x^{8} + 12x^{4}y^{5} + 9y^{10})$
Corrected: $(4x^{4} - 3y^{5})(16x^{8} + 12x^{4}y^{5} + 9y^{10})$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 174 (leaf n185), reads "16. (4x^4 - 3y^5)(16x^8 + 12x^4y^5 + 9y^10).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G009: Answers, Exercise 36, answer 1, p. 175
Printed: $(x + 1)(a^{2} - x + 1)$
Corrected: $(x + 1)(x^{2} - x + 1)$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 175 (leaf n186), reads "1. (x + 1)(x^2 - x + 1).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G010: Exercise 37, problem 18, p. 80
Printed: $m^{2}n^{2} + 14mnx^{2} + 49x^{2}$
Corrected: $m^{2}n^{2} + 14mnx^{2} + 49x^{4}$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 80 (leaf n91), reads "18. m^2n^2 + 14mnx^2 + 49x^4.", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G011: Answers, Exercise 39, answer 21, p. 176
Printed: $(x - 8)(x + 11)$
Corrected: $(x - 8)(x - 11)$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 176 (leaf n187), reads "21. (x - 8)(x - 11).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G012: Answers, Exercise 47, answer 4, p. 178
Printed: $\dfrac{4ax}{a^{3} - x^{2}}$
Corrected: $\dfrac{4ax}{a^{2} - x^{2}}$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 178 (leaf n189), reads "4. 4ax/(a^2 - x^2).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G013: Exercise 51, problem 11, p. 105
Printed: + 1 - \frac{1}{4} = 0
Corrected: + 1\frac{1}{4} = 0
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 105 (leaf n116), reads "11. (2x + 1)/4 - (4x - 1)/10 + 1 1/4 = 0 (a mixed number).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
G014: Answers, Exercise 76, answer 3, p. 183
Printed: $4frac{1}{6}$
Corrected: $4\frac{1}{6}$
transcription (the 1894 page is right; the 2011 transcription differs)
Confirmed against a scan of the printed book: the transcription differs, the book is right.
Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 183 (leaf n194), reads "3. 4 1/6 (the backslash is missing, so the transcription renders as 4frac16).", read from the page image on 2026-10-10. Flagged by our checking pipeline: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.
Scan: https://archive.org/details/bwb_P9-CSD-871
N001: Worked example 5 before Exercise 17 (Divide), p. 45
Printed: \Item{5.} Divide $-15a^{3}b^{2}c^{3}$~by~$-60a^{2}bc^{3}$.
Corrected:
note (no correction proposed)
A note on the text, not an erratum.
Status: the scan and the transcription differ; the 1894 page is itself inconsistent. The Internet Archive scan bwb_P9-CSD-871, printed page 45 (leaf n56), reads "5. Divide -15a^3b^2c^3 by -10a^2bc^3." with the working below it dividing by -60a^2bc^3 to get ab/4. So the 1894 printing contradicts itself, and the transcription reads -60 in both places, either from a later corrected printing or a silent correction. Recorded so a reader comparing a copy is not surprised; the transcription's -60 is the self-consistent reading.
Scan: https://archive.org/details/bwb_P9-CSD-871