## ALGEBRA

4/1/2015 CBSE

ALGEBRA

• The totality of all rational and all irrational numbers forms the set of
all real numbers. Thus every natural number, every whole number, every
integer, every rational number and every irrational number is a real number.

• The numbers of the form
where p and q are
integers and are known as rational numbers.

• Irrational numbers ��� All numbers which when expressed in decimal form or
in non terminating and non-repeating form, are known as irrational numbers.

• Every positive even integer is of the form 2q and that every positive odd
integer is of the form 2q+1, where q is some integer.

• A number greater than 1 is called a prime number, if it has exactly two
factors, namely 1 and itself.

• The prime factorization of a natural number is unique, except for the
order of its factors.

• If ���p��� is a prime and p divides ���a2��� then ���p��� divides ���q��� where ���a��� is a
positive integer.

• Numbers greater than 1 which are not prime, are known as Composite
numbers.

• If a number is divisible by ���p��� as well as ���q��� where p and q are
co-primes, then the given number is divisible by pq.

• If p(x) is a polynomial in x, the highest power of x in p(x) is called the
degree of the polynomial p(x)

• A polynomial of degree 1 is called a linear polynomial, while a polynomial
of degree 2 is called a quadratic polynomial.

• Every polynomial is a rational expression but a rational expression need
not be a polynomial.

• If p(x) and g(x) are any two polynomials with , then we can find
polynomials q(x) and r(x) such that ��� p(x) = g(x) x q(x) x r(x)

where r(x) =0 or degree of r(x) < degree of g(x)

• The zeroes of a polynomial p(x) are precisely the x-coordinates of the
points, where the graph of y=p(x) intersects the x-axis.

• A quadratic polynomial can have at most 2 zeroes and a cubic polynomial
can have at most 3 zeroes.

• An equation of the form ax + by = c, where a , b and c are real numbers is
called a linear equation in two variables x and y. The graph of a linear
equation ax + by=c is a straight line.

• A system consisting of two simultaneous linear equations is said to be
consistent, if it has at least one solution.

• A system of two simultaneous linear equations is said to be inconsistent,
if it has no solution at all.

• The equations of the type ax + by = c and kax + kby = kc are known as
dependent equations.

• If a pair of linear equations is given by��� a1x+b1y +
c1 = 0 and ��� a2x+b2y + c2 = 0,
then the following situations can arise:

(pair of linear
equations is consistent)

(pair of linear
equations is consistent)

(pair of linear
equations is consistent)

• The difference between the largest and the smallest observation is known
as the range of a data, while the number of observations falling in a
particular class is called the frequency.

• When classes are of the form 1-11, 11-21, 21-31 etc. the distribution is
said to be in continuous form.

• Mean of observations is the sum of the values of all the observations
divided by the total number of observations.

• A mode is that value among the observations which occurs most often, that
is, the value of the observation having the maximum frequency.

• The median is a measure of central tendency which gives the value of
middle-most observation in the data-

i) If n is odd, the median is ���

i) ii. If n is even, the median will be

• There is a empirical relationship between the tree measures of central
tendency

3 Median = Mode + 2 mean

• We mark the upper class limits along x-axis and cumulative frequencies
along y-axis. Plot these points and join them to get a curve, called an ogive.

• Histogram is a graphical representation of a frequency distribution in the
form of rectangles with class intervals as bases and the corresponding
frequencies as hights.

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