#2 Visual Proof of Pythagoras theorem

#1 Visual proof of Pythagoras theorem

Video Rationalising the binomial denominator

Learn to rationalise the denominator...
Watch this video

Video Rationalising the denominator

Learn how to rationalise the denominator...
Watch this video.

Rational numbers

Is 0 a rational number?
Yes, 0 is a rational number as it cane be expressed as 0/1, which is of the form p/q, q not equal to 0 and p & q are real numbers.


Is square root of 4 a rational number?
Yes, square root of 4 is equal to 2 and 2 can be written as 2/1, which is of the form p/q, q not equal to 0 and p & q are real numbers.

Hippasus of Metapontum

Hippasus of Metapontum

Hippasus of Metapontum b. c. 500 B.C. in Magna Graecia, was a Greek philosopher. He was a disciple of Pythagoras. To Hippasus (or Hippasos) is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.

Until Hippasus' discovery, the Pythagoreans preached that all numbers could be expressed as the ratio of integers. Despite the validity of his discovery, the Pythagoreans initially treated it as a kind of religious heresy and they either exiled or murdered Hippasus. Legend has it that the discovery was made at sea and that Hippasus' fellow Pythagoreans threw him overboard.

Representing an irrational number on a number line Part(2)

Observe the given figure and write the procedure of representing √5 on the number line.

Representing an irrational number on a number line (Part 1)

Learn how to represent √2 on a number line.
1. First of all draw the number line.
2. Mark point A at "0" and B at "1". This means AB = 1 Unit.
3. Now, at B, draw BX perpendicular to AB.
4. Cut off BC = 1 Unit.
5. Join AC.
6. By Pythagoras theorem in right triangle ABC, we get AC = √2 Units.
7. Now, with radius AC and centre A, mark a point on the number line.
Let the marked point is M. M represents √2 on the number line.

Fact Sheet Real Numbers Class 10

Remember the following facts

1. Euclid’s division lemma:

Given positive integers a and b, there exist whole numbers q and r satisfying

a = bq + r, 0 ≤r < b.

2. The Fundamental Theorem of Arithmetic:

Every composite number can be expressed as a product of primes, and this

factorisation is unique, apart from the order in which the prime factors occur.

3. If p is a prime and p divides a², then p divides a, where a is a positive integer.

4. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are co-prime, and the prime factorisation of q is of the form2ⁿ5^m, where n, m are non-negative integers.

5. Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

6. Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2ⁿ 5^m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

Assignment 5- Number system class 9

Read the following statements and state True or False. Justify your answer with an example.
1.There are exactly 2 integers between 3 and 6.
2.There are exactly 2 natural numbers between 3 and 6.
3.There can be infinite natural numbers between two natural numbers.
4.There are exactly 2 rational numbers between 6 and 9.
5.There is one whole number which is not a natural number.
6.All natural numbers, whole numbers and integers are rational numbers.
7.A real number is either rational or irrational.
8.We cannot plot irrational numbers on a number line.

Euclid's Division Algorithm- Jug filling Problem

Watch this video and write your observations

Video- How to find HCF using Euclid's division Algorithm?

Video-Euclid's Division Algorithm

Video-The Euclidean Algorithm

Video- Euclid's Division Lemma

Euclid's Division Lemma

Video- Rational and Irrational numbers

Watch this video and recapitulate the concept of rational and irrational numbers. After watching this video make a list of 20 rational numbers and 20 irrational numbers in your notebook.

Density of Rational Numbers

What is a rational number?
A number which can be expressed in the form p/q, q not equal to zero, p and q both are integers is called a rational number. Examples: 2/3, -3/2, 5/7 etc.
All integers, natural numbers and whole numbers can also be expressed as a rational number.
An interesting thing to note about rational numbers is, that between any two rational numbers we can insert a rational number. Isn't it amazing?
This means, between any two rational numbers there exist infinite number of rational numbers.
Let us learn a method to insert a rational number between two given rational numbers. Consider two rational numbers say a and b. Can you find the middle most number between a and b? Yes, you are right. If you find the average of these two numbers, this will give you the middle most number say (c). This means, if I repeat the process, then I can find a number between a and c as well as a number between b and c. That's really interesting. So, keep on finding as many numbers between a and b using this method. In general, we say if a and b are two rational numbers then (a+b)/2 is a rational number between a and b.
Exercise:
1. Find two rational numbers between 1/2 and 1/3.
2. Find six rational numbers between 1/6 and 1/7.

We know that the decimal representation of a rational number is either terminating or non terminating and repeating.

Insert 5 rational numbers between 1.2 and 1.3

How will you do this?

Let a = 1.2 and b = 1.3

We can write them as a = 1.20 and b= 1.30
So, 5 rational numbers between 1.2 and 1.3 will be
1.21, 1.22, 1.23, 1.24 and 1.25
Now, insert 5 rational numbers between 1.20 and 1.21
How will you do this?

Video Density of Rational Numbers

Watch this video and recapitulate the concept of inserting rational numbers between two rational numbers.

Real number system-Types of numbers

Basic Concepts

1. Natural numbers- Counting numbers 1,2,3,4.....


2. Whole numbers- Numbers 0,1,2,3,4......


3. Integers- Numbers .............. -3,-2, -1, 0, 1, 2, 3, ............


4. Rational numbers- Numbers which can be expressed in the form p/q, q not equal to zero, p and q both are integers


5. Irrational numbers- Numbers which cannot be expressed in the form p/q, q not equal to zero, p and q both are integers

Observe the relation between various types of numbers in the diagram given above.

Important observations:

1. Rational numbers and Irrational numbers together forms a collection of Real numbers.


2. All natural numbers are rational numbers but not vice versa.


3. All whole numbers are rational numbers but not vice versa.


4. All integers are rational numbers but not vice versa.

Excercise:

Give examples to justify the above observations.

Syllabus Class 10 Mathematics 2011-12

Dear students,
Please note the term wise syllabus of class 10 Mathematics 2011-12.

Syllabus Class 10 Mathematics 2011-12 -

Syllabus Class 9 Mathematics 2011-12

Dear students,
Please note down the syllabus for both the terms.

Syllabus Class 9 Mathematics 2011-12