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Number and Operations are an essential part of the study of Mathematics. A number is a quantity that is used in counting and measuring. The definition of the term number includes such numbers as zero, negative numbers, rational numbers and complex numbers. The study of Number and Operations involves understanding representations, relationships and number systems
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Homework help and unlimited tutoring in Number Theory
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Number theory gets easy with TutorVista's unlimited tutoring and homework help. Get expert help in understanding Number theory proofs, Complex numbers, Basic Number theory and Introduction to Number theory and Indices.
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Rational and Irrational Numbers |
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The rational numbers can be expressed as terminating or recurring decimals. |
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Numbers which are not rational are called irrational numbers. When expressed as decimals, they are non-terminating and non-recurring.
We can obtain infinite number of irrationals between two irrational numbers. |
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The union of the set of rational numbers and irrational numbers forms the set of real numbers.
Problems involving rational numbers are simplified using 'BODMAS' rule. |
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Significant Figures |
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We use specific number of digits to denote an exact value of a number for required accuracy. The digits used for such a purpose are called significant figures. |
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The difference between these two values is called the absolute error.
Absolute Error = Original number - Approximated number
The ratio of the absolute error and the original number expressed in percentage form is called percentage relative error. |
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Sets |
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In mathematics, a well-defined collection of definite objects is called a set. |
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George Cantor is regarded as the "Father of Set theory". |
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The concept of "Sets" is basic in all branches of mathematics.
Important terms in Sets are: Roster Method, Set builder form, Finite and Infinite Sets, Null Set or Empty Set or Void Set, Singleton Set or Singlets, Equivalent Sets, Equal Sets, Cardinality of a Set, Universal Set, Subsets, Power Set. |
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Indices |
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If m is a positive integer, a x a x a ... m times is written as am. a is called the base and m is the power. We read it as "a raised to the power m". The power is also called "the index" or "the exponent".
An irrational root of a positive rational number is called a surd. |
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Polynomials |
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An algebraic expression of the form a0+a1x+a2x2+….+anxn where |
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a0, a1, a2,….an are real numbers, n is a positive integer is called a polynomial in x. |
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Complex Numbers |
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Square root of a negative number is known as an imaginary number. |
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If x and y are real numbers, then x + iy is called a complex number.
x is called the real part and y is called the imaginary part. |
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The following are the types of complex numbers: Equality of Complex numbers, Sum of two Complex numbers, Negative of a Complex number, Additive identity of the Complex number, Additive inverse of a Complex number, Product of two Complex numbers, Multiplicative identity of Complex numbers, Conjugate complex numbers, Quotient of two non-zero Complex numbers, Reciprocal of a non-zero complex number or multiplicative inverse of a non-zero complex number. |
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Matrices |
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1. A matrix is a rectangular array of numbers, arranged in rows and columns. |
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2. Order of a matrix = Number of rows in it x Number of columns in it |
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3. Row matrix is a matrix with only one row. |
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4. Column matrix is a matrix with only one column. |
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5. Zero or Null matrix is a matrix in which every element is zero. |
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Matrices and Determinants |
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Matrices : A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. |
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Determinants : Let A = [aij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or |A|. |
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Sequences and Series |
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Sequence : A set of numbers arranged in a definite order according to some definite rule is called a sequence.A sequence is a function whose domain is the set N of natural numbers.
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Series : Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series. |
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Mathematical Induction |
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The word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a principle by which one can arrive at a conclusion about a statement for all positive integers, after proving certain related proposition. |
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Binomial Theorem |
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1. A sentence is called a statement if it can be adjudged as true or false. Every statement is a sentence, but a sentence may or may not be a statement. |
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2. A statement involving natural number n is generally denoted by P(n). |
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3. A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'. |
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Ratio and Proportion I |
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Ratio: If a and b are two quantities with the same units such that b is not equal to zero; then the quotient a/b is called the ratio between a and b. It is written as a:b.
Ratio does not have any unit.
The quantities a and b are called terms of the ratio.
a' is called antecedent (first term)
b' is called consequent (second term).
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Proportion: Four quantities a, b, c and d are said to be in proportion, if a:b=c:d. a and d are called extremes of the proportion. b and c are called means of the proportion.
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Ratio and Proportion II |
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Ratio is the numerical relationship between two quantities of the same kind. The first quantity is called the antecedent and the second quantity is called the consequent. |
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By performing simple operations on ratios, we get compounded ratio, duplicate ratio, triplicate ratio, sub-duplicate ratio and sub-triplicate ratio. |
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If two ratios are equal, we get four quantities in proportion. |
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If three quantities are in continued proportion, the second quantity is called the mean proportion. |
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Important results in proportions are: Invertendo, Alternendo, Componendo, Dividendo and Componendo and Dividendo.
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Question and Answers 1 |
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Question and Answers 2 |
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Question and Answers 3 |
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Question and Answers 4 |
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