Rational and Irrational Numbers
The rational numbers can be expressed as terminating or recurring decimals.
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.
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.
Natural Numbers
The countable numbers are called Natural Numbers.
Therefore 1 is the smallest or least of all natural numbers.
The highest natural number cannot be said because Natural numbers are infinite.
The set of Natural numbers is a subset of Rational numbers
Whole numbers
The set of whole numbers is the set of natural numbers along with zero. so W = the set of whole numbers = 0,1,2,3,............
so Zero is the least number of the set of Whole numbers.
As the whole numbers is an infinite set we cant determine the highest number of this set.
The set of Whole numbers is a subset of Rational numbers
Significant Figures
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.
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.
Divisibility, Divisors or Factors and Multiples,
If a number has a whole number answer when divided by a second number, the first number is divisible by the second number; x is divisible by y if and only if x = qy where y is a whole number.
Prime numbers and Composite numbers
A prime number is one that has only two factors namely 1 and itself.
and a composite number has factors besides 1 and itself.
Greatest Common Factor and Least Common Multiple
The greatest of the common divisors of two or more than two numbers is called the GCD or HCF (Highest Common Factor).
The least common multiple of two or more than two numbers is called the LCM
Fractions
Fraction is an equal part of one whole object.
Fraction can be represented as " p/q " where 'p' denotes the value called numerator and 'q' denotes the value called denominator.
Decimals
Every fraction with denominator 10 can be written in decimal notation.
1/10 can be written as 0.1 in decimal notation.The dot represents the decimal point and it come between the units place and the tenths place. One will read this as zero point one.
Percentage
Percent means 'per -cent' -> per hundred. It is the way of expressing any given value in terms of hundreds. which is denoted by '%' symbol.
Sets
In mathematics, a well-defined collection of definite objects is called a set.
George Cantor is regarded as the "Father of Set theory".
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.
Unit of Conversions
Conversion of units refers to conversion factors between different units of measurement for the same quantity.
Indices
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.
Scientific Notation
Notation of large number in terms of exponential form, 10 raised to required value. It is also a standart notation representation as " p x 10q ". where 'p' and 'q' are any integers.
Polynomials
An algebraic expression of the form a0+a1x+a2x2+….+anxn where
a0, a1, a2,….an are real numbers, n is a positive integer is called a polynomial in x.
Complex Numbers
Square root of a negative number is known as an imaginary number.
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.
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.
Matrices
1. A matrix is a rectangular array of numbers, arranged in rows and columns.
2. Order of a matrix = Number of rows in it x Number of columns in it
3. Row matrix is a matrix with only one row.
4. Column matrix is a matrix with only one column.
5. Zero or Null matrix is a matrix in which every element is zero.
Matrices and Determinants
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.
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|.
Logarithms
If a > 0 such that ay = x then y is called the logarithm of x with respect to the = base ‘a’ and written as logax = y.
Sequences and Series
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.
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.
Mathematical Induction
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.
Binomial Theorem
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.
2. A statement involving natural number n is generally denoted by P(n).
3. A binomial is an algebraic expression of two terms which are connected by the operations '+' or '-'.
Ratio and Proportion I
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).
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.
Ratio and Proportion II
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.
By performing simple operations on ratios, we get compounded ratio, duplicate ratio, triplicate ratio, sub-duplicate ratio and sub-triplicate ratio.
If two ratios are equal, we get four quantities in proportion.
If three quantities are in continued proportion, the second quantity is called the mean proportion.
Important results in proportions are: Invertendo, Alternendo, Componendo, Dividendo and Componendo and Dividendo.
