Number System - Edu Foundation

Number System

If we talk about Number System,
Then it is same as talking about Real Numbers.
Real Numbers are nothing but a bag of all the numbers that everyone uses all over the world.

Note:- All Integers and fractions are rational numbers.

But Why we consider this?

Because they are written in p/q form.

So, Rational numbers are numbers which are written in p/q form and q is not equal to 0.

Ex. 2/3, 4, -2, 3/9 etc.

What is p and q here?

p and q are numerator and denominator respectively.

So What about Irrational Numbers?

If rational numbers are written in p/q form.

Then Irrational Numbers must be opposite of that.

So Irrational Numbers are not written in p/q form and q is not equal to 0.

Ex. √3, √5 etc.

After knowing about the facts,

Come to the first concept -

Rational Numbers

How to find rational numbers between two given numbers?

If we assume to find rational numbers between -3/11 and 8/11.

Then answer is easy to find

-> -2/11,-1/11,0/11,1/11,2/11,3/11,4/11,5/11,6/11,7/11

But if we assume to find rational numbers between-3/13 and 9/13

Then we have to multiply 10 on both Nr and Dr of both numbers.

After that, we get -30/130 and 90/130

Then the answer is like this

-> -29/130,-28/130,-27/130,-26/130,-25/130,-24/130,-23/130,-22/130,-21/30,-20/130

Then how to find decimal representation of rational numbers in p/q form.

Simply use long division method to find the solution.

See below if 8/3 is given as question.

If a decimal representation is given and asks us to find it in p/q form

So what we do!

Before going to that concept,

We have to aware about the types of rational numbers.

So what do you mean by the first type and second type?

Ist type means if we divide a rational number by another rational number, then the quotient is a exact number either it may be a number or a decimal.

2nd type means if we divide a rational number by another rational number then the quotient is the number which is repeated. It means that the division never ends and repeated digits in the decimal are placed under a bar.

So come to the concept that we left,

If a terminating decimal representation is given like 837/1000,83/100,8/10000

The anwer will be 

837/100 - 0.837

83/100 - 0.83

8/10000 - 0.0008

It is easy to find as it needs the method of power of tens.

But if a non terminating and repeating decimal representation is given

Then,

Why bars are placed above digits after the decimal point?

This signifies that the digits which are placed under the bar are repeated continuously.

The Denominator (Dr) depends on numbers under the bar.

If two numbers present under the bar.

Then, we put two nines in the Dr.

If there is three, then put three in Dr,like this.

Another type of non terminating and repeating decimal representation is given like that-

See question (I) not (ii) as it is half captured.

 You have to carefully attend this type of questions.

Irrational Numbers

 As we all study about the meaning of this number in the previous post. Also take examples √3, √5, 0.1010010001....... But Why we consider √3 and √5 as irrational but not rational. There is two proves behind it,

1. Analysing Prove -

This is nothing but we have to carefully look at the numbers in order to find whether it is terminating or non terminating repeating or non terminating non repeating. Finally we consider that irrational numbers will have non terminating non repeating decimal representation because it never ends and do not repeated.

2. Contradiction Prove -

See below -

Note:- Some useful results on Irrational Numbers (IMP 1.1) :-

1. Negative of an irrational number is an irrational number.

2. The sum of a rational number an irrational number is an irrational number.

3. The product of a non zero rational number and an irrational number is an irrational number.

4. The sum, difference, product, and quotient of two irrational numbers need not be an irrational number.


This is all about Irrational Numbers. So after knowing all the concepts of Rational and Irrational Numbers, we have to discuss about the questions from the topic that we are going to appear in upcoming exams.

1. How to identify √45 as rational or irrational?

Assume √45
√45 = √[5×9]
       = √[5×3×3]
       = 3√5
Here 3 is rational and √5 is an irrational.
As we have already studied that the product of a non zero rational number and an irrational number is an irrational number. (See IMP 1.1)
So, √45 is an irrational number.

2. How to insert a rational number and an irrational number between 2 and 3?

Before doing that, we have to know about two formulas.

Formula 1 for finding a rational number between two given number:- a+b/2 ( where a  and b are the first and second numbers respectively)

Formula 2 for finding an irrational number between two given number:- √ab

So come to the question that we left before,

rational number between 2 and 3 is a+b/2 = 2+3/2 =5/2

irrational number between 2 and 3 is √ab = √(2×3) = √6
So these are the required answers.

How to prove that (√2+2)² is an irrational number?
Answer given below:-

Thus (√2+2) is an irrational number.

How to find two irrational numbers for which the product of both is a rational or an irrational?
There is no formula or theorems required to find it, Only you have to guess carefully and assuming two numbers.
For example:-
If I assume one number as √2 and another number as √3, then the product of both is √2 ×√3 = √6 (irrational).
If I assume one number as √3 and another number as also √3×√3 = 3 (rational).

How to do graphical representation of irrational numbers on the number line?

In order to represent √2 on it, we have to follow the algorithm given below:-

Step 1 - Draw any line m
Step 2 - Place points on the line such as o, A, A1, A2, A3 and so on, such that the distance between each of them is 1 unit.
Step 3 - Draw a ray by taking A as centre, perpendicular to line m assuming that the length of the line formed in 1 unit and named it AB.
Step 4 - Then join O and B to get a right triangle OAB.
Step 5 - By taking O as centre, Draw an arc of measure OB which cut the line at A1. This A1 point represents √2 on the line.

From where I get √2?
Nothing but the measure of OB.
By using Pythagoras Theorem, we can easily find it.
(OB)²=(OA)²+(AB)²
(OB)²=(1)² +(1)²
(OB)²=1+1
(OB)²=2
OB=√2 units
In this way, we represent an irrational number on the number line.
In case, it is given to represent √3, √5 in the line?
After representing √2, Take B as centre and draw another perpendicular line assuming the length of it is 1 unit.
Similarly follow th upe same steps as √2 to represent √3 on the number line.
Similarly, in order to represent √5, we have to assume OA as 2 units and follow the same steps as above.

How to locate square root of a decimal point on the number line?

Step 1 - Obtain the positive real number x (say)

Step 2 - Draw a line and Mark a point A on it.

Step 3 - Mark a point B on the line such that AB = x units.

Step 4 - From point B mark a distance of 1 unit and Mark the new point as C.

Step 5 - Find the midpoint of AC and Mark the point as O.

Step 6 - Draw a circle with centre O and radius OC.

Step 7 - Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Length BD is equal to √x.

Why BD is equal to√x?

We have to justify it.

Justification:- 

How to visualise Real Numbers on the number line by using the successive Magnification?

It is one of the most easiest method of representation of real numbers on number line.

Assume to locate 4.2626

See below - 

Draw a number line containing all integers.

At first, we have to see that between which numbers, 4.2626 is located.

That is 4 and 5.

Then draw another number line containing all integers between 4 and 5 like - 4.1, 4.2, 4.3,......, 4.9.

Similarly we have to see the next digit that is 2.

Then we have to see between which numbers 4.26 is located, that is 4.20 and 4.30.

Then draw the next number line as given in the picture above.

You have to continue this process until we have not get the the number 4.2626.

Now we are going to study about Class 10 First chapter Real numbers which is a part from Number System. This chapter includes only two properties of positive integers,
1. Euclid's Division Algorithm
2. Fundamental Theorem of Arithmetic

       Euclid's Division Algorithm 

Euclid's Division Algorithm otherwise called Euclid's division lemma which says that any positive integer a can be divided by another positive integer in such a way that it leaves a remainder r that is smaller than b.

So the actual theorem is - 

Given positive integers a and b, There exist unique integers q and r satisfying a = bq +r (0<_r < b).

Note:-

1. It is stated for only positive integers.

2. It can be extended for all integers except zero.

Steps for doing Division Lemma:-

1. Apply Euclid's division lemma, to c and d. So, we find whole numbers, q and r such that c = dq +r, 0 <_r < d.

2. If r = 0, d is the HCF of c and d. If r is not equal to 0, apply the division lemma to d and r.
3. Continue this process till the remainder is zero. The divisor at this stage will be the required HCF.
This algorithm works because HCF (c,d) = HCF (d,r) where the symbol HCF (c,d) denotes the HCF of c and d, etc.

By the above property of positive integers, you will have to appear such questions in the upcoming examinations:-

1. How to find HCF of 210 and 55 by using Euclid's division Algorithm?
-> Since 210 > 55
So, By applying division lemma to 210 and 55, we get:-
210 = 55 × 3 + 545
Again by applying division lemma to the new divisor 55 and new remainder 45, we get:-
55 = 45 × 1 + 10
Again by applying division lemma to the new divisor 45 and new remainder 10, we get:-
45 = 10 × 4 + 5
Again by applying division lemma to the new divisor 10 and new remainder 5, we get:-
10 = 5 × 2 + 0
Here the remainder at this stage is zero. So the procedure stops.
Finally the divisor 5 is the HCF of 210 and 55.

If the question says us to find the HCF of three numbers, So what we do ?
Just follow the steps given below:-
1. Find the HCF of any two of the given numbers.
2. Find the HCF of the third given number and the HCF obtained in step 1.
3. The HCF obtained in step 2 is the HCF of three given numbers.

Then LCM,
Everyone knows how to find LCM, So I omits that.
But in which cases, you do HCF and LCM?
1. If there is maximum, then you have to find HCF and if there is minimum, then you have to find LCM.
2. If it asked for needs, or grouping same things, Then you have to find HCF.
3. HCF of two numbers is a divisor of LCM of same two numbers.

Then come to the second property of positive integers.

Fundamental Theorem of Arithmetic 

Fundamental theorem of arithmetic is nothing but something that works with the multiplication of positive integers.  It says that 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.

Actual meaning of Fundamental Theorem of Arithmetic :-

Here both are same,
But arrangement of digits are different.
So, Fundamental theorem of arithmetic always do it in an unique way.

Formula to remember:-
HCF (a,b) × LCM (a,b) = a × b
It states that If a is a number and b is another number, Then its product is equal to the product of LCM and HCF of that same numbers.

Note:- Above formula is applicable, if two numbers are given but not three or more.
It means that if a and b are given, Then it is possible.
But if a, b and c are given, Then it is not.
LCM (a,b,c) × HCF (a,b,c) is not equal to a × b × c

To state whether a rational number will have a terminating representation or a non - terminating representation, we have to do long division process which we studied in class 9.
But in class 10, A new method for finding it, is easier than the long division method.

Method:-
If x is a rational number equal to p/q and prime factorisation of q is in the form 2 power n and 5 power m, or 2 power n, or 5 power m, where m and n are non - negative integers. Then x has a decimal representation which terminates.
You can easily unserstand it by the given examples:-

How to prove If p divides a2, then p divides a?

Watch the video:-

If you have still any doubts on the concepts, Then feel free to comment us.