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Madam : Archana ,Can You Pick the Integers from the Circle ?
Archana : It's very easy mam. 2 and 5
Madam : Whatever you picked are integers that's fine but you
missed a couple more.
Archana : Is it so mam but I can't find anymore integers here.
Madam : Why don't you consider zero and negative integers as integer too. -1 and 0 both are integers.
Archana : What are the other numbers are they called as non-integers ?
Madam : Definitely the other numbers are non integers but they don't fall in the same category.
Archana : It's very easy .Let me do it.
7/2 = 3.5
-3/5 = -0.6
1/3 = 0.3333........ It's a non terminating value.
Root 2 = 1.41421356.....It's a non terminating as well as non-recurring value.
Madam : Now you understand why I told you not to put the other numbers in the same category.
The terminating ones are called as Terminating Rational number. e.g 7/2 , -3/5
The non-terminating and repeating or recurring one is called as
Non-Terminating Rational number .e.g 1/3
How to express a non terminating decimal ?
1/3 = 0.3333... = 0.3 (put a dot over 3)
9/110 = 0.0818181... = 0.81(put a dot over 8 and 1) or (draw a line over 8 and 1)
Put a dot(.) above first and last repeating digit or draw a line above the repeating digits.
I know what are you going to ask me next .
What is a rational number ?
Any number that can be expressed in the form of m/n where m ,n are integers
and n not equals to 0 is called as Rational Number.
Is 5 a rational number ?
Yes because 5 can be expressed as 5/1 so here m=5 and n=1
Here we can see m and n are integers and n not equals to zero so 5 is a
Rational Number .
The number which can not be expressed as Rational Number are called as
Irrational Number
- Few theorems on number systems
and P is a prime number.
2. If ( I1 x I2) is divisible by P then either ( I1 or I2 is divisible by P) or both (I1 and I2 are
divisible by P) where I1,I2 are non-zero integers and P is a prime number.
3 If I to the power n is divisible by P then I is divisible by P too where I is a non-zero
integer , n is a positive integer and P is a prime number .
Is root 2 a rational number ?
The answer is let us assume root 2 as a rational number and check whether it satisfies
the properties of a rational number or not.
Check the proof in the video tutorial below.
Surds
We have already discussed about irrational numbers like root 2, root 3 but still we missed some of the
irrational numbers like cube root of 2 , cube root of 7 ,etc - -- these numbers can't be expresses as cubes of any rational number.Such numbers are called as surds.
So n th root is a surd if
1.a is a positive rational number.
2. n > 1 and n is an integer.
3. nth root of a is not a rational number.
We can represent
- nth root of a = a to the power of 1/n
- (nth root of a) to the power of n = a
- (nth root of a) (nth root of b) = nth root of (ab)
- (nth root of a)/(nth root of b) = nth root of (a/b)
where n>1 a,b are positive rational number.
Rationalisation
Multiplying a surd with another surd to make it a rational number is called as rationalisation.
Example :
multiply (5 + root3) with (5 - root3) to make it arational number (5square -3)=23
This is called rationalisation.
Representation real numbers in a Line
Representation of real numbers in a line are given in the video tutorial below
great video ...
ReplyDeletethis link talking about irrational number too
http://www.math-worksheets.co.uk/048-tmd-what-are-irrational-numbers/