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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to **nub****trek**.

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

**Just keep tapping** (or clicking) on the content to continue in the trail and learn. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about. continue

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen. continue

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge. continue

At some values, Functions may evaluate to an indeterminate value. At those values, will the function

(a) remain indeterminate?

(b) continuous?

To answer these understand limit of a function. *(click for the list of lessons in this topic)*

Basics: Limit of a function

A function can take the form `0/0` or discontinuous. In this, the basics are explained to identify if, at an input value, a function is defined or continuous.

A function can take the form 0 by 0 or discontinuous. In this, the basics are explained to identify if, at an input value, a function is defined or continuous.

The pages in this lesson are

• Indeterminate value and Undefined Large

• Indeterminate value in functions

• Expected Value of a Function

Understanding limits with Graphs

The geometrical meaning of left-hand-limit and right-hand-limit are explained with graph of a function. The function is considered as two constituent functions of numerator and denominator and using the graphs of these functions, the limit is explained.

The geometrical meaning of left-hand-limit and right-hand-limit are explained with graph of a function. The function is considered as two constituent functions of numerator and denominator and using the graphs of these functions, the limit is explained.

The pages in this lesson are

• Understanding limits with the graph of the function

• Understanding limits with Graphs of Numerator and Denominator

Calculating Limits

Learn how to examine a function at an input value. Based on the information, how to determine if the function is defined, continuous, or not defined.

Learn how to examine a function at an input value. Based on the information, how to determine if the function is defined, continuous, or not defined.

The pages in this lesson are

• Examining Function at an input value

• Limit of continuous functions

• Limit of piecewise functions

Algebra of Limits

If a function is given as sum or multiplication of multiple functions, then how to find limit of the function? Algebra is limit is about rules to applying limits to sub-expressions of the function.

If a function is given as sum or multiplication of multiple functions, then how to find limit of the function? Algebra is limit is about rules to applying limits to sub-expressions of the function.

The pages in this lesson are

• Algebra of Limits : Introduction

• Limit of Sum or Difference of functions

• Limit of Quotient of functions.

Limit of Algebraic Expressions

Standard results for limits of function involving polynomials and evaluating to `0/0`, `oo/oo`, or `oo - oo` are examined and explained with examples.

Standard results for limits of function involving polynomials and evaluating to 0 by 0 , infinity by infinity, or infinity minus infinity ; are examined and explained with examples.

Limit of Trigonometric, Logarithmic, Exponential Functions

Standard results for limits of Trigonometric, Logarithmic, and Exponential functions are explained with diagrams.

Standard results for limits of Trigonometric, Logarithmic, and Exponential functions are explained with diagrams.

The pages in this lesson are

• Limit of Trigonometric Functions

How to handle quantities that have magnitude and direction? Vectors are a convenient way to represent such quantities. Start this topic to understand the definition, applications, and properties of vectors. *(click for the list of lessons in this topic)*

Introduction to Vector Quantities

A person walks 3m east and then, 4m north. From that position, what is the distance she has to walk back to the starting point?

The answer is not 3m + 4m. Some quantities do not add in amount, and require direction to be specified. The vector algebra deals with representation and arithmetic of such quantities.

A person walks 3 meter east and then, 4 meter north. From that position, what is the distance she has to walk back to the starting point?;; The answer is not 3 meter + 4 meter. Some quantities do not add in amount, and require direction to be specified. The vector algebra deals with representation and arithmetic of such quantities.

The pages in this lesson are

Properties of Vectors

This topic discusses magnitude of a vector and the properties of that. This also discusses the different types of vectors like null, proper, collinear, coplanar, etc.

This topic discusses magnitude of a vector and the properties of that. This also discusses the different types of vectors like null, proper, collinear, coplanar, et cetera.

The pages in this lesson are

Vectors and Coordinate Geometry

Coordinate Geometry is the system of geometry where the position of points on the 3D coordinates are described as ordered pairs of numbers corresponding to the three axes. In the mathematical representation of vectors, the components along 3D coordinates are described individually. In this topic, you will learn how these two are related.

Coordinate Geometry is the system of geometry where the position of points on the 3D coordinates are described as ordered pairs of numbers corresponding to the three axes. In the mathematical representation of vectors, the components along 3D coordinates are described individually. In this topic, you will learn how these two are related.

Direction - Unique Feature of Vectors

In this topic, you will learn about the basic mathematical operations using vectors, and how direction of vectors affect the definition of the mathematical operations.

Starting on learning "". ;;

The pages in this lesson are

Vector Addition

In this topic, you will learn how the vectors are added. Two insights into vector addition : sequential addition and continuous addition, are explained.

Starting on learning "Vector Addition". ;; In this topic, you will learn how the vectors are added. Two insights into vector addition : sequential addition and continuous addition, are explained.

The pages in this lesson are

• Vector Addition - First principles

Properties of Vector Addition

Vector addition is an extension of addition of real numbers as individual components. The properties of vector addition can be easily understood with the properties of real number addition.

Starting on learning "Properties of Vector Addition". ;; Vector addition is an extension of addition of real numbers as individual components. The properties of vector addition can be easily understood with the properties of real number addition.

The pages in this lesson are

• Properties of vector addition

• Closure Property of Vector Addition

• Commutative Property of Vector Addition

Multiplication of Vectors by Scalar

In this topic, multiplication of vectors by scalar is explained as repeated addition of vectors. This is used in the component form of vector representation with three unit vectors.

Starting on learning "Multiplication of Vectors by Scalar". ;; In this topic, multiplication of vectors by scalar is explained as repeated addition of vectors. This is used in the component form of vector representation with three unit vectors.

Properties of Vector Multiplication by Scalar

This topic explains properties of multiplication of vector by scalar with examples.

Starting on learning "Properties of Vector Multiplication by Scalar". ;; This topic explains properties of multiplication of vector by scalar with examples.

The pages in this lesson are

• Applicable Properties and understanding them

• Order of Scalar Multiplication

• Distributive over Scalar Addition

• Distributive over Vector Addition

• Multiplication by Product of Scalars

Vector Dot Product

Do two vectors interact to form a product? Vector dot product is defined for components in parallel.

Starting on learning "vector dot product". ;; Do two vectors interact to form a product? Vector dot product is defined for components in parallel.

The pages in this lesson are

• Vector Dot Product : First Principles

• Vector Dot Product : Projection of a vector

Properties of Dot Product

This topic explains properties of vector dot product with examples.

Starting on learning "Properties of Dot Product". ;; This topic explains properties of vector dot product with examples.

The pages in this lesson are

• Understanding Properties of Dot Product

• Dot product of a scalar multiple

• Dot product with a null vector

• Dot product of Orthogonal Vectors

• Dot product of Collinear Vectors

• Distributive Property over Vector Addition

• When products of two vectors are equal

• Dot Product : No Multiplicative Identity

Vector Cross Product

Do vectors interact to form a product? Vector cross product is defined as the product of components in perpendicular.

Starting on learning "Vector Cross Product". ;; Do vectors interact to form a product? Vector cross product is defined as the product of components in perpendicular.

The pages in this lesson are

• Introduction to Cross Product

• Cross Product: First Principles

• Cross Product : Area of Parallelogram

Properties of Cross Product

This topic explains properties of vector cross product with examples.

Starting on learning "Properties of Cross Product". ;; This topic explains properties of vector cross product with examples.

The pages in this lesson are

• Understanding Properties of Cross Product

• Cross Product : Closure Property

• Cross Product : Not Commutative

• Cross product of a Scalar Multiple

• Cross product with a null vector

• Cross Product of Orthogonal Vectors

• Cross Product of Collinear Vectors

• Distributive Property over Vector Addition

• When products of two vectors are equal

• Cross Product : No Multiplicative Identity

What are trigonometric ratios `sin`, `cos`, `tan`? What are the practical applications for these ratios? This lesson series explains the trigonometric ratios defined for triangles, standard angles, and basic trigonometric identities. *(click for the list of lessons in this topic)*

Basics of Trigonometry

In this lesson, the basics required to understand trigonometry are revised. Then, one application scenario for sides of right angles triangle is illustrated. With that, the trigonometric ratios are defined for right angled triangles.

This series on trigonometry is split into 5 lessons and you are starting on the first lesson. ;; In this lesson, the basics required to understand trigonometry are revised. Then, one application scenario for sides of right angles triangle is illustrated. With that, the trigonometric ratios are defined for right angled triangles.

Trigonometric Ratios for Standard Angles

In this lesson why some angles are chosen as standard angle is explained and then the method to calculate trigonometric ratios of standard angles is explained.

Starting on learning "Trigonometric Ratios for Standard Angles". ;; In this lesson why some angles are chosen as standard angle is explained and then the method to calculate trigonometric ratios of standard angles is explained.

Trigonometric Basic Identities

The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"

Starting on learning "Triogometric Identities". ;; The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"

What is the the solution for `x` in the equation `x^2=-1`?

To find solutions to similar problems, real number system is extended to *complex number system*.

In this topic, learn the definition, applications, and properties of complex numbers. *(click for the list of lessons in this topic)*

Introduction to Complex Numbers

Irrational numbers were introduced as solution to algebraic equations similar to `x^2=2`. Similarly, complex numbers are introduced as *solutions to algebraic equations* similar to `(x-a)^2=-b`.

At the end of this topic, the *generic form of complex numbers* is proven to be `a+ib`.

Irrational numbers were introduced as solution to algebraic equations similar to x squared equals 2. Similarly, complex numbers are introduced as solutions to algebraic equations similar to x minus a squared equals minus b. At the end of this topic, the generic form of complex numbers is proven to be a + i b.

The pages in this lesson are

• Number System Quick Revision

• Representation of Complex Numbers (incomplete)

Complex Plane and Polar Form

Complex number `a+i b` is equivalently an ordered pair `(a,b)` which can be abstracted to represent a 2D plane. This is named after the mathematician JR Argand as *Argand Plane* or *complex plane* with real and imaginary axes.

Complex numbers, having abstracted to a complex plane, are represented in polar form.

Complex number a+i b is equivalently an ordered pair; a,b ; which can be abstracted to represent a 2D plane. This is named after the mathematician J R Argand as Argand Plane, or, complex plane, with real and imaginary axes. ;; Complex numbers, having abstracted to a complex plane, are represented in polar form.

The pages in this lesson are

• Complex Plane or Argand Plane

Algebra of Complex Numbers

Algebra of complex numbers details out the operations addition, subtraction, multiplication , division, conjugate, and exponent.

The algebra of complex numbers is quite easy to understand in abstraction. This topic goes beyond and explains the application context.

Algebra of complex numbers details out the operations addition, subtraction, multiplication , division, conjugate, and exponent. ;; The algebra of complex numbers is quite easy to understand in abstraction. This topic goes beyond and explains the application context.

The pages in this lesson are

• Complex Number: Modeling sine waves

• Understanding Complex Arithmetic

• Addition of two Complex numbers

• Subtraction of Complex Numbers

• Multiplication of two Complex numbers

• Conjugate of a complex Number

• Exponent of a complex Number

Properties of Complex Number Arithmetic

Complex number system is an extension of Real number system with inclusion of a new number `i=sqrt(-1)`. The complex arithmetic has the * properties mostly identical* to real arithmetic. The additional features are related to modulus, argument, and conjugate.

Complex number system is an extension of Real number system, with inclusion of a new number i or square root of minus 1. The complex arithmetic has the properties mostly identical to real arithmetic. The additional features are related to modulus, argument, and conjugate.

The pages in this lesson are

• Understanding Properties of Complex Arithmetic

• Multiplication : Closure Law

• Multiplication : Commutative Law

• Multiplication : Associative Law

A quantity `y` is specified as a function of a variable, and another quantity is derived to be rate of change of `y`. Derivatives provide the mathematical framework to derive and analyze the rate of change of the given function. *(click for the list of lessons in this topic)*

Introduction to Differential Calculus

Some quantities are related such that one is rate of change of another. Finding the rate of change of a quantity, as a function of variable, is the derivative or differentiation. In this, finding derivative in first principles, graphical meaning of derivatives, and the differentiability of function are explained.

Starting on learning "Introduction to Differential Calculus". Some quantities are related such that one is rate of change of another. Finding the rate of change of a quantity, as a function of variable, is the derivative or differentiation. In this, finding derivative in first principles, graphical meaning of derivatives, and the differentiability of function are explained.

The pages in this lesson are

• Differential Calculus : Understanding Application Scenarios

• Differentiation: First Principles

Algebra of Differentiation

In this page, finding derivatives for functions in various forms is explained.

Starting on "Algebra of Differentiation". In this page, finding derivatives for functions in various forms is explained.

The pages in this lesson are

• Understanding Algebra of Derivatives

Standard Results in Derivatives

In this page, standard results in derivative, that are repeatedly used, are explained and proven.

Starting on "Standard Results in Derivatives". In this page, standard results in derivative, that are repeatedly used, are explained and proven.

The pages in this lesson are

• Derivatives of Algebraic Expressions

• Derivatives of Trigonometric Functions

A quantity `y` is specified as a function of a variable, and another quantity is derived to be aggregate of change of `y`. Integrals provide the mathematical framework to derive and analyze the aggregate of change of the given function. *(click for the list of lessons in this topic)*

Basics of Integration

Some quantities are related such that one is aggregate of change of another. Finding the aggregate of change of a quantity, as a function of variable, is the integral or integration. In this, finding integral in first principles, graphical meaning of integrals, indefinite integrals, definite integrals, and fundamental theorem of calculus are explained.

Some quantities are related such that one is aggregate of change of another. Finding the aggregate of change of a quantity, as a function of variable, is the integral or integration. In this, finding integral in first principles, graphical meaning of integrals, indefinite integrals, definite integrals, and fundamental theorem of calculus are explained.

The pages in this lesson are

• Integral Calculus: Understanding Application Scenarios

• Integration: First Principles

• Integration: Graphical Meaning

Algebra of Integrals

In this page, finding integrals for functions in various forms is explained.

Starting on learning "Algebra of Integrals". In this page, finding integrals for functions in various forms is explained.

The pages in this lesson are

• Understanding Algebra of Integration

• Properties of Indefinite Integrals

Various Forms and Results of Integrals

In this page, several methods are explained to work out integrals of standard forms of functions.

Starting on learning "Various Forms and Results of Integrals". In this page, several methods are explained to work out integrals of standard forms of functions.

The pages in this lesson are

• Integration using Identities

Trigonometric ratios are introduced as ratio of sides of right angled triangles. In such a case, the angle can be in the range 0 to 90 degree. In advanced trigonometry, the definition is extended to unit circle form and trigonometric values are defined for all angles. *(click for the list of lessons in this topic)*

Trigonometric Values for all angles in unit circle form

The trigonometric values in unit circle form and Computing trigonometric values for an angle in any quadrant are explained.*Keep tapping on the content to continue learning.*

Started on learning "Trigonometric values for all angles in unit circle form". ;; The trigonometric values in unit circle form and Computing trigonometric values for an angle in any quadrant are explained.

The pages in this lesson are

• Trigonometric Values: Unit Circle Form

Trigonometric Ratio for Angles in All Quadrants

In this lesson, representing trigonometric values for angles in all quadrants of coordinate plane to an equivalent trigonometric value in first quadrant is explained.*Keep tapping on the content to continue learning.*

In this lesson, representing trigonometric values for angles in all quadrants of coordinate plane to an equivalent trigonometric value in first quadrant is explained.

The pages in this lesson are

Trigonometric Identities for compound angles

In this page, expressing trigonometric values for sum or difference of two angles, in terms of the trigonometric functions of the two angles is explained.*Keep tapping on the content to continue learning.*

In this page, expressing trigonometric values for sum or difference of two angles, in terms of the trigonometric functions of the two angles is explained.

The pages in this lesson are

• Compound Angles: Geometrical Proof for `sin(A+B)`

• Compound Angles: cos(A+B), sin(A-B), cos(A-B)

Numbers are used to count objects. And, sometimes, objects are divided or split into pieces. How are the pieces counted along with the whole objects? Fractions are introduced to count part of whole. For this new class of numbers, the comparison, addition, subtraction, multiplication, and division are explained. *(click for the list of lessons in this topic)*

Fractions : Fundamentals

Fractions are numbers with varying place value, that is, a part in a whole. A fraction is represented as a number and the place value together. This page explains the fundamentals of fraction.

Starting on learning "fundamentals of Fractions". ;; Fractions are numbers with varying place value, that is, a part in a whole. A fraction is represented as a number and the place value together. This page explains the fundamentals of fraction.

The pages in this lesson are

Types of fractions

Based on the properties, the fractions are classified as different types : like fractions, unlike fractions, proper fractions, improper fractions, mixed fractions, equivalent fractions.

Based on the properties, the fractions are classified as different types : like fractions, unlike fractions, proper fractions, improper fractions, mixed fractions, equivalent fractions.

The pages in this lesson are

• Proper, Improper, and Mixed fractions

Arithmetics with Fractions

Fractions are specified in different place values or denominators. This page explains arithmetics with fractions such as, comparison, addition, subtraction, multiplication, and division.

Started on learning "Comparison of Fractions". Fractions are specified in different place values or denominators. This page explains arithmetics with fractions such as, comparison, addition, subtraction, multiplication, and division

The pages in this lesson are

Numerical expression is a quantity represented with numbers and arithmetic operations between them. The numbers in such numerical expression can be replaced with varying quantities, called variables. The expressions with variables are called algebraic expressions. Study of such algebraic expressions is algebra. *This title introduces algebra at 6th to 8th grade level **(click for the list of lessons in this topic)*

Algebra: Variables and Expressions

Numbers are standard symbols representing constant values of quantities. In place of numbers, any symbol can be used to represent a varying value. Such symbols are variables. Variables are used in algebraic expressions and algebraic equations.

Starting on learning "Variables and Expressions in algebra". Numbers are standard symbols representing constant values of quantities. In place of numbers, any symbol can be used to represent a varying value. Such symbols are variables. Variables are used in algebraic expressions and algebraic equations.

The pages in this lesson are

Polynomials - Basics

Polynomials are introduced as simplified algebraic expressions. The classification of polynomials based on number of terms, number of variables, and power of variables are discussed.

Starting on learning about "Polynomials". Polynomials are introduced as simplified algebraic expressions. The classification of polynomials based on number of terms, number of variables, and power of variables are discussed.

The pages in this lesson are

Polynomial Arithmetics

Like numbers or numerical expressions, polynomials can be added, subtracted, multiplied or divided.

Starting on learning "Polynomial Arithmetics". Like numbers or numerical expressions, polynomials can be added, subtracted, multiplied or divided.

The pages in this lesson are

• Addition and Subtraction of Polynomials

Basic Algebraic Identities

Identities are statements equating two expressions. In this, standard algebraic identities are introduced and explained.

Identities are statements equating two expressions. In this, standard algebraic identities are introduced and explained.

The pages in this lesson are

• Algebraic Identities of Square