math4u

Tuesday, May 15, 2012

Euclids Division Lemma

Euclid's division lemma states that " For any two positive integers a and b, there exist integers q and r such that a=bq+r , 0≤r<b.

If we divide a/b,then we get q as the quotient and r as the remainder such that it satisfies a=bq+r where

r,remainder is greater than or equal to zero

b,divisor is always greater than the remainder r

When the divisor b is perfectly divisible then remainder r is zero,if not divisible remainder is less than divisor

a-dividend

b - divisor

q - quotient

r-remainder

Dividend =divisor*quotient + remainder

consider the examples

22/7

Dividing 22/7 gives 3 as quotient and 1 as remainder

22-dividend

7 - divisor

3- quotient

1-remainder

Here 7 is greater than 1,so we can write 22/7 as

22=7*3 + 1 ,0≤ 1< 7

81/9

81=9*9, here remainder is zero ,ie 0<9

58/5

58=5*11+ 3 , 0≤ 3< 5

In all the above cases r is greater than or equal to zero and less than b.

Real numbers

Real numbers

All numbers on the number line or the numbers which exists..Real numbers are usually denoted by the letter R.

Real numbers contain:

(1, 2, 3, ...) natural numbers or counting numbers denoted by N
(0, 1, 2, 3, ...) whole numbers denoted by W
(-3,-2, -1, 0, 1, 2,3, ...) integers denoted by Z
rational numbers (p/q, where p and q are any two numbers (except that q cannot be 0))
irrational numbers (such as √2 or π)
square roots,cube roots etc

Every natural numbers are whole numbers.Every whole numbers are integers .Every integers are rational numbers and every rational numbers are real numbers. So real number is the combination of rational numbers and irrational numbers

Whole number = natural number and zero

Integer = whole numbers and negatives

Rational number = integer/integer(we can make any integer a rational number by putting it over 1)

Example: 0, -9 ,6/3 ,π,24,or 0.333333 .......

Rational Numbers

The numbers that can be written as the ratio o f two integers are called rational numbers or fractions.

Rational numbers

can be represented as simple fraction
both numerator and denominator are integers and denominator cannot be zero
the result of a fraction is a terminating or repeating decimal.
can be put on a number line.

Examples

2.5 is rational, because it can be written as the ratio 5/2

78 is rational, because it can be written as the ratio 78/1

3.333... (3 repeating) is also rational, because it can be written as the ratio 10/3

sqr(25) is also rational since it can be written as 5/1

-60/3 is rational ,because it can be written as -20/1

An integer is a rational number since any integer can be made as a fraction by dividing it with 1.
But a rational number always may not be an integer

Example: 10/1 is an integer, but 58/3 is not

Irrational Numbers

These are the numbers that cannot be expressed as a simple fraction or ratio.

Irrational numbers

real numbers
cannot be represented as integer/integer
non repeating, non terminating decimal

Examples

sqr(2)=1.414213........

π = 3.141592..........

In both the above example,they cannot be written as a fraction and also the digits after the decimal are non repeating and non terminating,

A number can either be rational or irrational, but not both.

An irrational number is either non-terminating or non-repeating but rational number terminates or repeates. There is no overlap between these two number types.

Examples

0.25

This is a terminating decimal, so it can be written as a fraction: 25/100 = 1/4.

3.141592653...

Here the decimal does not repeat, so this is an irrational. the answer is: irrational

3.1415

This is a terminating number. The answer is: rational, real

Find whether the numbers are rational or irrational

58/9 ,7 8/3 ,–sqrt(64) ,sqr(3)

58/9 is a fraction ,so rational

7 8/3 = (3*7+8)/3 = 29/3,so rational

-sqr(64) = -8, is an integer.so rational

sqr(3) = 1.732050............,so irrational

Monday, May 14, 2012

Syllabus

UNIT I : NUMBER SYSTEMS

1. REAL NUMBERS

Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work

done earlier and after illustrating and motivating through examples, Proofs of results -irrationality of decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

UNIT II : ALGEBRA

1. POLYNOMIALS

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.

Statement and simple problems on division algorithm for polynomials with real coefficients.

2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Pair of linear equations in two variables and their graphical solution. Geometric representation of

different possibilities of solutions/inconsistency.Algebraic conditions for number of solutions. Solution of a pair of linear equations in two variables algebraically - by substitution, by elimination and by cross multiplication.

UNIT III : GEOMETRY

1. TRIANGLES

Definitions, examples, counter examples of similar triangles.

1. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

2. If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.

3. If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.

4. If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.

5. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.

6. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

7. The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.

8.In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

9. In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.

UNIT IV: TRIGONOMETRY

1. INTRODUCTION TO TRIGONOMETRY

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well

defined); motivate the ratios, whichever are defined at . Values (with proofs) of the trignometric

ratios. Relationships between the ratios.

2. TRIGONOMETRIC IDENTITIES

Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given.

Trigonometric ratios of complementary angles.

UNIT VII: STATISTICS AND PROBABILITY

1. STATISTICS
Mean, median and mode of grouped data (bimodal situation to be avoided).Cumulative

frequency graph.

3. QUADRATIC EQUATIONS

Standard form of a quadratic equation .Solution of the

quadratic equations (only real roots) by factorization, by completing the square and by using

quadratic formula. Relationship between discriminant and nature of roots.

4. ARITHMETIC PROGRESSIONS

Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first

n terms and their application in solving daily life problems.

UNIT III: GEOMETRY (Contd.)

2. CIRCLES

Tangents to a circle motivated by chords drawn from points coming closer and closer to the

point.

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of

contact.

2. (Prove) The lengths of tangents drawn from an external point to circle are equal.

3. CONSTRUCTIONS

1. Division of a line segment in a given ratio (internally)

2. Tangent to a circle from a point outside it.

3. Construction of a triangle similar to a given triangle.

UNIT IV: TRIGONOMETRY

3. HEIGHTS AND DISTANCES

Simple and believable problems on heights and distances. Problems should not involve more

than two right triangles.

UNIT V : STATISTICS AND PROBABILITY

2. PROBABILITY

Classical definition of probability. Connection with probability as given in Class IX. Simple

problems on single events, not using set notation.

UNIT VI : COORDINATE GEOMETRY

1. LINES (In two-dimensions)

Review the concepts of coordinate geometry done earlier including graphs of linear equations.

Awareness of geometrical representation of quadratic polynomials. Distance between two points

and section formula (internal). Area of a triangle.

UNIT VII : MENSURATION

1. AREAS RELATED TO CIRCLES

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas

and perimeter / circumference of the above said plane figures. (In calculating area of segment of

a circle, problems should be restricted to central angle of only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

2. SURFACE AREAS AND VOLUMES

(i) Problems on finding surface areas and volumes of combinations of any two of the following:

cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.

(ii) Problems involving converting one type of metallic solid into another and other mixed

problems.

Tuesday, September 27, 2011

UNIT I : NUMBER SYSTEMS

1. REAL NUMBERS

Euclid's division lemma, Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of results -irrationality of decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.