Number line

Number line are the line made up with numbers. A line is actually made up with points. Eventually distributed points are called line. So the each point in a  number line have a particular number. that is why we are saying the number line is made up with numbers.

After drawing a line just mark any one point as 0. Then if we mark a particular unit of distance as number 1 in the number line, then the point for the number 2 will come exact same distance from number one. Like this each and every points in a number line have a particular number.

Now we will just start to walk through number line from zero. And we will pick all the numbers like 1,2,3 etc.. Then the collection of all this number can called natural numbers. Denoted by (N).

Now we have to walk back all the way and pick the zro also. So now we have the collection of natural number and zero. And it can be called as whole numbers (w). Whole numbers are 0,1,2,3  etc….. . Now again we walk back we will see lot of negative integer numbers. And we will collect al that numbers also. now we have the collection of positive integer number , zero and negative integer numbers.  The collection of all this number together can call  integer numbers (Z).

Is any numbers left in the number line?. Yes the numbers like 1/2, 3/4,  are left in the number line. So we will collect that numbers also.Including the numbers like 1/2, 3/4 with naturel number is called rational number (R) . Here we can say one definition for rational number.

Any number which can be write in the form of  P/Q is called rational numbers.

Here P  and Q are integers and Q not equal to zero.

Here we cn notice that all natural numbers are rational as per this definition. Because all the natural numbers can be write with denominator 1.

Zero is also a rational number because it can be write with denominator as 1.

And the negative integers also rational number because it can also write with denominator 1. So The numbers so far we studied are all rational numbers.

How ever we now that a number in the form of P/Q is not unique. Example 1/2=2/4= 25/50=47/94. Because these all are equal numbers (fractions.). So when ever we mark a rational number in the form of P/Q then we have to make the number in the lowest form ppf fraction. Or the P and Q have only one as common factor (co-prime numbers).

Are the following are true or false give your reason

1. Every whole number is natural numbers.
2. Every integer is rational numbers.
3. Every rational number is an integer.
Finding rational numbers between two rational numbers

If we have given two rational numbers, then we can find the rational number between them by find the mean value. We have to add the given two numbers and then divided by 2. Then we will get another rational number between them. by repeating this method we can find infinity of rational numbers between two rational numbers.

Example : find the two rational number between 1 and two.

(1+2)/2=3/2 is a rational number between 1 and 2.

(2+3/2)/2 = 4/2+3/2= (7/2 )/2 =7/4 is an another rational number between 1 and 2.

Another method

Suppose  we want to find 5 numbers between 1 and two at a single step, like follows.

As we need 5 numbers we have to make  denominator as 6 for both number. Here

1= 6/6 and 2 = 12/6

Now we can write 7/6, 8/6, 9/6, 10/6, 11/6. these all are in between 6/6 and 12/6 and hence between 1 and 2.

Notes: Also remember that there will be infinite number of rational numbers between any two rational numbers.

Do the exercise 1.1

Irrational numbers

As we agai check on the number line still there is lot of numbers left in between rational numbers. So now we will see how that numbers is different from rational numbers.

If a number is irrational then it cannot be written in the form of p/q, where p and q are integers and q not equal to zero.

Some examples are √2, √3, √15, π, 0.123400678…… .

There will be infinite number of irrational numbers in the number line.

After collecting all the irrational numbers from the number line, then the number line will empty as there is any more number. So the collection of all these numbers we studies so for together can called the real number. (R).

The proof for √2 as an irrational number will study in the nest class. But now we will study how we can mark the √2 in a number line.

Locate √2 on the number line

Consider the unit square OABC. As shown in the fig.1.7. Now consider the diagonal OB. And we can consider the triangle OAB, it is aright angle triangle. Therefore we can say that

OB= OA2 +AB2

OB=√OA2 +AB2

OB= √1+1 =√2

Now consider the vertex O of the square are coincide with 0 of the number line. And the OB which is equal to √2 can now marked on the number line by draw an arc with radius OB. The ar will intersect on the number line at the point P And the distance from 0 to p is √2.

Locate √3 on the number line

This can also mark on the number as same as √2. For that first we have to draw one more right angle triangle in the previous fig. One side of this right angle triangle is OB (√2) and other side DB is unit length and OD is the hypotenuse. see the below fig.

real number and their decimal expansion

A we study the real number having rational number and irrational numbers. In this section we will analyze the decimal expansion of both type of number and understand the difference of the rational number and irrational number.

First we will check the decimal expansion of three different rational number.  They are 10/3, 7/8, 1/7. See the below fig for there division.

In the first case  10/3 we can see that the quotient is not ending. But it is repeating as a block . This type of decimal expansion can call non terminating recurring. It can write as 3.3(bar). That means the decimal places are repeating with 3,3,3,3..

In the second case 7/8 we can see that the quotient is terminating after a finite number. This type of decimal expansion can call terminating decimal expansion.

In the third case 1/7 the quotient is not ending but it is also repeating as block of  142857, 142857,…… so it can write as 0. 142857(bar). So this can also call non terminating recurring decimal expansion.

Now we have to proof the above decimals are rational numbers by wring in the form of P/Q . As P and Q are integers. it should be a perfect number ( cannot be repeated or require in their decimal expansion).

First we will consider a terminating decimal number 3.142678.  Here the decimal places are 6 digits. Therefore we can multiply and divide the 3.142678 with 1000000, so that it will become 3142678/1000000 so now it is in the form of p/q  and hence it is a rational number.

In this second case we will show that the non terminating recurring number is a rational number, by writing the number as p/q.

consider 0.3(bar), here 0.3333… we don’t know the exact number. so we will take

x= 0.333333….  ( we can consider multiplying x with 10 because the nuber have 1 digit requiring number)

10x= 10 *( 0.33333…..) = 3.333333….

3.3333….=3+x (since x= 0.3333…) (here the obtained number after multiplication will write as sum of a perfect number and the given number x)

10x=3+x

9x=3 and x=3/9= 1/3 this is in the form p/q. and Hence it is a rational number.

Let us try with other number 1.272727 her 27 is repeating.

Here also we don’t know the exact number so we will take

x= 1.2727…..

100x = 100 * 1.2727… =127.2727…

100x =126+x (here the obtained number after multiplication will write as sum of a perfect number and the given number x)

99x=126 so x=126/99. which is 14/11 so it is also a rational number.

From the above studies we can understand the decimal number with terminating decimal places and the number with non terminating recurring decimal places are rational number.

Irrational number

From the above studies we can say that any number is irrational if it cannot be written in the form of p/q. Where p and q are integers and q not equal to 0.

In the decimal expansion we see the terminating and non terminating recurring decimal number are rational numbers. Because they can write in the form of p/q.

But some decimal number like 0.1011001111000001 are not terminating and not requiring. this type of decimals cannot be written in the form of p/q. And hence it is not rational number. ex. √2, Π.

Finding irrational numbers.

Finding irrational number between 1/7 and 2/7. First find the value of 1/7 which is 0.142857(bar). And the value of 2/7 = 0.285714(bar).

Now easily we can write the irrational number in between them ex.

0.156578911112. we can write such infinity of irrational number between two. By this same method we can write irrational numbers between any two real numbers.

representing real number on the number line

We can represent any real number in a number line. Suppose we ant to mark 5.3775 (5 decimal places) as the number is between 5 and 6 we will make the portion of the number line with 5 and 6. Then we will divide that portion in 10 equal parts. The we can mark the first decimal digit 5.3. A our number is greater than 5.3 and less than 5.4 we will draw anew portion of number line with 5.3 and 5.4. this process can repeat until we mark the complete given number. Please see the below fig for the details.

Do some exercises for practice.

Arithmetical operation on real numbers

I earlier class we studied the rational numbers are satisfy the commutative, associative and distributive laws for addition and multiplication. If we add, subtract,  multiply and divide (Except zero) two rational numbers we will get a a rational number . So rational number are closer.

The irrational number also tisfy the commutative, associative and distributive laws for addition and multiplication.

How ever the sume , difference, product and quotient are not always irrational.

ex. (√6)+(-√6), (√2)-(√2), (√3)*(√3),(√17)/(√17)

By doing some exercise we can summarize following  points.

1. The sum or difference of a rational number and an irrational number is irrational.
2. The product or quotient of a non zero rational number with an irrational number is irrational.
3. If we add, subtract, multiply or divide two irrationals, then the result may be rational or irrational.
Geometrical method to find the square root of a positive real number

Finding the √3.5 geometrically.

Procedure: Draw a line AB with 3.5 unit length. From B extend one more unit length and mark C. Now find the midpoint of AC and mark that point as O. Now draw the semicircle with the radius AO. A s shown in the figure. Draw a perpendicular passing through B and intersec the semi circle. And then the length of BD= √3.5

Proof : Not required for grade 9

We can also use the above method to mark root of any number on the number line. See the below figure, in that the BC is the unit length And we can extend the line with more unit lent and mark B AS 0, C as 1 and so on… Now we can draw an arc with center B and radius BD, then it will intersect on the number lane.

Identities for roots

The above are the 6 identity for roots. By this identity we can do various mathematical operation on roots. Do some exercises for more exposure.

Laws of exponent

I addition we can also say that  nth root of a= a1/n.

And  am/n = nthroot of am

That means if a number have the power with fractional number, then the denominator of the fraction will become the power of the root and the numerator will the power of all thing.

Practice some exercises