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I am looking for books which are not just typical math. books by just giving least theory and difficult exercise. I am looking for a book that develops the real thinking ability and along with it covers problem soling skills by providing interesting questions. If you know about a book covering only one or two chapters of the following list, I will still love to read that.





Fundamental principle of counting. Permutations and Combinations, derivation of

formulae and their connections and simple applications.



Principle of Mathematical Induction and its simple applications.


Binomial theorem for positive integral indices, general term and middle term,

properties of Binomial coefficients and simple applications.




Arithmetic and Geometric progressions, insertion of arithmetic, geometric means

between two given numbers. Relation between A.M. and G.M. Sum upto n terms of

special series n, n2, n3. Arithmetico - Geometric sequence.



Trigonometric functions. Trigonometrical identities and equations. Inverse

Trigonometric functions, their properties and applications.


Complex numbers as ordered pairs of reals. Representation of complex numbers in

a plane. Argand plane and polar representation of complex numbers. Algebra of

complex numbers, modulus and argument (or amplitude) of a complex number,

square root of a complex number, triangle inequality. Quadratic equations in real

and complex number system and their solutions. Relation between roots and coefficients,

nature of roots, formation of quadratic equations with given roots.




Sets and their representations. Union, intersection and complement of sets and their

algebraic properties. Power Set. Relation, types of relations and equivalence

relation. One-one, into and onto functions and composition of functions. Real -

valued functions, algebra of functions, polynomials, rational, trigonometric,

logarithmic and exponential functions, inverse functions. Graphs of simple

functions. Even and odd functions.



Limit and continuity of a function, limit and continuity of the sum, difference, product

and quotient of two functions, L'Hospital rule of evaluation of limits of functions.

Differentiability of functions. Differentiation of the sum, difference, product and

quotient of two functions. Differentiation of trigonometric, inverse trigonometric,

logarithmic, exponential, composite and implicit functions; derivatives of order up

to two. Rolle’s and Lagrange’s Mean Value Theorems. Applications of derivatives:

rate of change of quantities, monotonic - increasing and decreasing functions,

maxima and minima of functions of one variable, tangents and normals.




Integral as an anti-derivative. Fundamental integrals involving algebraic,

trigonometric, exponential and logarithmic functions. Integration by substitution,

by parts and by partial fractions. Integration using trigonometric identities. Definite

Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite

integrals. Evaluation of definite integrals. Applications of the integrals:

determining areas of the regions bounded by simple curves in standard form.



Ordinary differential equations, their order and degree. Formation of differential

equation whose general solution is given. Solution of differential equations by the

method of separation of variables. Solution of homogeneous differential equations

and linear first order differential equations.



Cartesian coordinate system, distance formula, section formula, locus and its

equation, translation of axes, slope of a line, parallel and perpendicular lines,

intercepts of a line on the coordinate axes.

Straight lines : Various forms of equations of a line, intersection of lines, angles

between two lines, conditions for concurrence of three lines, distance of a point

from a line, equations of internal and external bisectors of angles between two lines,

coordinates of centroid, orthocentre and circumcentre of a triangle, equation of

family of lines passing through the point of intersection of two lines.

Circles, Conic sections : Standard equation of a circle, general form of the equation

of a circle, its radius and centre, equation of a circle when the end points of a

diameter are given, points of intersection of a line and a circle with the centre at the

origin and condition for a line to be tangent to a circle, equation of the tangent.

Sections of a cone, standard equations and properties of conic sections (parabola,

ellipse and hyperbola), condition for y = mx + c to be a tangent and point (s) of




Coordinates of a point in space, distance between two points, section formula.

Direction ratios and direction cosines of a line joining two points, angle between


two intersecting lines. Coplanar and Skew lines, the shortest distance between two

lines. Equations of a line and a plane in different forms, intersection of a line and a




Scalars and vectors, addition of vectors, components of a vector in two dimensional

and three dimensional spaces, scalar and vector products scalar and vector triple




Matrices, algebra of matrices, types of matrices, elementary row and column

operations. Determinant of matrices of order two and three. Properties of

determinants, area of triangles using determinants. Adjoint and inverse of a square

matrix. Test of consistency and solution of system of linear equations in two or

three variables using inverse of a matrix.



Measures of Dispersion: Calculation of mean, median, mode of grouped and

ungrouped data. Calculation of standard deviation, variance and mean deviation

for grouped and ungrouped data.

Probability: Probability of an event, addition and multiplication theorems of

probability, Baye’s theorem, probability distribution of a random variable,

Bernoulli trials and Binomial distribution.


Any help is highly appreciated!

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As a reference book I keep a copy of the Handbook of Mathematics by Bronshtein-Semendyayev. It has no instructional material or exercises but it covers everything you've listed above.

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As a reference book I keep a copy of the Handbook of Mathematics by Bronshtein-Semendyayev. It has no instructional material or exercises but it covers everything you've listed above.


I am not looking for reference books currently but full text books type. But, Thanks!

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Basic Set Theory - Azriel Levy


This book is both completely incomprehensible and completely intuitive at the exact same time! Exercises are interspersed between propositions, lemmas, and corollaries in a very argument driven outline that gives a step by step account of the subject leading up to paper machines.



Differential Equations and Linear Algebra - Edwards & Penney


Excellent text book that incorporates the two topics respectively. The questions in this book are challenging but are formulated in such a way that the concepts become relevant to their application.

Edited by Xittenn
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  • 3 weeks later...

Here are a few titles that spring to mind, though most are admittedly ones I haven't read, instead being included based on recommendations from many others more experienced than I. I'll add more over time, probably.


General mathematics:


Proof and logic:




I also have and somewhat enjoy a few of Debra Anne Ross' Master Math series, though they're more reference books than textbooks.

Edited by John
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Has anyone used the three volumes on Calculus by Marsden and Weinstein? Here's volume one.


No, but Marsden and Weinstein are very accomplished mathematicians. I have used R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Addison–Wesley (1987) which I highly recommend to anyone interested in geometric mechanics.

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  • 2 weeks later...

I would have just added this to my previous post, but for whatever reason (maybe browser compatibility) I'm unable to edit the thing. I just came across this:




which might be worth browsing. It's a collection of books recommended for learning various mathematics.



I just ordered Rudin's three books on analysis today, one of which is on that list. They've been recommended reading to me here, on another forum, and on freenode! I'm sure they will be the win . . . .


Principles of Mathematical Analysis - Walter Rudin


Real and Complex Analysis - Walter Rudin


Functional Analysis - Walter Rudin


and Topology - Munkres was everything it was said to be as well . . . .

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  • 1 year later...

I apologize for the thread necromancy, but this thread is still on the first page of its section at least. smile.png


Recently, I found out about this: http://people.math.gatech.edu/~cain/textbooks/onlinebooks.html


It's a collection of links to math textbooks freely available online. Obviously I haven't read most of them, but at least a few that I briefly checked out seem decent. A few of the links are broken, but in general it's a pretty nice list.

Edited by John
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