CBSE NOTES CLASS 9 MATHEMATICS CHATER 1

NUMBER SYSTEMS

SUMMARY OF BASIC CONCEPTS

N = Natural numbers = {1,2,3, ….}

W = Whole numbers = {0,1,2,3, ….}

Z = Integers = {….,-3,-2,-1,0,1,2,3, ….}

Q = Rational numbers

R = Real numbers,

“A number ‘r’ is called a rational number, if it can be written in the form r = $\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are co-prime integers and q ≠ 0. (Why do we insist that q ≠ 0?)”

• The real numbers which are not rational numbers (i.e. which cannot be represented in $\frac{\mathrm{p}}{\mathrm{q}}$ form are called irrational numbers.

• All rational numbers and irrational numbers, examples, 0.123, 0, 1, -3, $\frac{3}{4}$, π, etc.

• All natural numbers are also whole numbers

• All natural numbers and whole numbers are also integers.

• All natural numbers, whole numbers and integers are also rational numbers.

• All natural numbers, whole numbers, integers and rational numbers are also real numbers.

• Real numbers includes all rational numbers and all irrational numbers.

• Represent number ; q > p; on the number line.

• Locating irrational numbers on the number line$\sqrt{2}$ $,\sqrt{3}$ etc.

• Real numbers and their decimal expansions

• Decimal expansion of a rational number is either terminating or non-terminating recurring, while the decimal expansion of an irrational number is non-terminating non-recurring.

• Representing Decimal Exapansion of Real Numbers on the Number Line - Process of Successive Magnification

Operations on Real Numbers

• The sum or difference of a rational number and an irrational number is irrational.

• The product or quotient of a non-zero rational number with an irrational number is irrational.

• If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.

Idenities for exponents of real numbers

For positive real numbers a and b,

 3. 4. 5. 6.

Let a > 0 be a real number and p and q be rational numbers. Then

To rationalize the denominator of we multiply $\frac{1}{\sqrt{\mathrm{a}}}$ by $\frac{\sqrt{\mathrm{a}}}{\sqrt{\mathrm{a}}}$where, a is an integer.

To rationalize the denominator of we multiply where a and b are integers.

CBSE NOTES CLASS 9 MATHEMATICS CHATER 1

NUMBER SYSTEMS

DETAILED CHAPTER NOTES

N = Natural numbers = {1,2,3, ….}

W = Whole numbers = {0,1,2,3, ….}

Z = Integers = {….,-3,-2,-1,0,1,2,3, ….}

Q = Rational numbers

R = Real numbers,

“A number ‘r’ is called a rational number, if it can be written in the form r = $\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are co-prime integers and q ≠ 0. (Why do we insist that q ≠ 0?)”

Examples, $\frac{1}{2}$, $\frac{3}{4}$, 0.1, 0.4 1, 0, -4, etc.

Co-prime numbers are those numbers which do not have common factors, except 1. 2, 9 are co-prime. 3 and 9 are not co-prime.

• The real numbers which are not rational numbers (i.e. which cannot be represented in $\frac{\mathrm{p}}{\mathrm{q}}$ form are called irrational numbers.

• All rational numbers and irrational numbers, examples, 0.123, 0, 1, -3, $\frac{3}{4}$, π, etc.

• All natural numbers are also whole numbers

• All natural numbers and whole numbers are also integers.

• All natural numbers, whole numbers and integers are also rational numbers.

• All natural numbers, whole numbers, integers and rational numbers are also real numbers.

• Real numbers includes all rational numbers and all irrational numbers.

Example 1: Are the following statements true or false? Give reasons for your answers.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

[NCERT]

Solution:

(i) False, because zero is a whole number but not a natural number.

(ii) True, because every integer m can be expressed in the form $\frac{\mathrm{m}}{1}$, and so it is a rational number.

(iii) False, because $\frac{3}{5}$ is not an integer

Number Line

Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.

This is why we call the number line, the real number line.

Represent number $\mathbit{}\frac{\mathbit{p}}{\mathbit{q}}$; q > p; on the number line.

Divide the one whole in the q parts equal to denominator and mark the whole number as $\frac{\mathrm{q}}{\mathrm{q}}$. Each part represents $\frac{1}{\mathrm{q}}$.

For example 1$\frac{3}{4}$ can be represented as follows,

• There are infinite natural numbers, infinite whole numbers, infinite integers, infinite rational numbers, infinite irrational numbers and infinite real numbers.

• There are infinite rational, infinite irrational and infinite real numbers between any two numbers.

• Square root of any real number, which is not a perfect square, is irrational, e.g. … etc.

• Finding n rational numbers between two rational numbers a and b.

Multiply a and b both by

$\frac{\mathrm{n}+1}{\mathrm{n}+1}$

For example, find 5 numbers between $\frac{1}{4}$ and $\frac{1}{2}$ Make the denominators of both the numbers equal.

Multiply both numbers by $\frac{6}{6}$, then the numbers will become, $\frac{6}{24}$ and $\frac{12}{24}$, then the numbers, $\frac{7}{24}$, $\frac{8}{24}$, etc. are rational numbers lying between $\frac{1}{4}$ and $\frac{1}{2}$.

We can also multiply both the numbers by $\frac{10}{10}$, $\frac{100}{100}$ etc.

Example 2: Find five rational numbers between 1 and 2.

[NCERT]

Solution:

Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1,

• Another way is to find the midpoint of both the numbers. This number lies between both the numbers, and continue doing it.

• We can generate infinite number of irrational numbers, for example 0.101100111000…. etc.

Example 3: Locate $\sqrt{2}$ on the number line.

Solution:

Draw OA = 1 unit. Draw a perpendicular AB at A, such that AB = 1 unit. Now by Pythagoras theorem,

Now taking O as centre and OB as radius, draw an arc, which cuts the number line at P. point P now represents $\sqrt{2}$

Example 4: Locate $\sqrt{3}$ on the number line.

Solutions:

Construct BD of unit length perpendicular to OB. Then using the Pythagoras theorem, we see that

Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point Q.

Then Q corresponds to $\sqrt{3}$.

• In the same way, you can locate $\sqrt{\mathrm{n}}$ for any positive integer n, after $\sqrt{\mathrm{n}-1}$ has been located.

Decimal Expansion

• The decimal expansion of a rational number can be terminating or nonterminating recurring.

• A number whose decimal expansion is terminating or non-terminating recurring is rational.

• The decimal expansion of an irrational number is non-terminating non-recurring.

• A number whose decimal expansion is non-terminating non-recurring is irrational.

Example 5: Find the decimal expansions of $\frac{10}{3}$, $\frac{7}{8}$ and $\frac{1}{7}$

Solution: Try yourself using long division method.

Rational Numbers with Repeating Decimal Expansion

We represent repeating digits in a rational number under a bar. For example 1.2727…. can be represented as $1.\stackrel{̅}{27}$

• The maximum number of digits in the repeating block of digits in the decimal expansion of $\frac{1}{\mathrm{n}}$ can be n-1.

Example 6: Show that 3.142678 is a rational number. In other words, express 3.142678 in the form $\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are integers and q ≠ 0

Example 7: Represent $1.\stackrel{̅}{27}$ in the form of $\frac{\mathrm{p}}{\mathrm{q}}$.

Solution:

Let x = $1.\stackrel{̅}{27}$ = 1.2727…. (1)

Multiply the number by 10, 100, 1000 etc., depending upon the number of repeating digits.

100x = 127.2727…. (2)

Subtracting equation (1) from (2), we get,

Example 9: Show that 0.2353535... = 0.235 can be expressed in the form $\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are integers and q ≠ 0.

Solution:

= 0.23535… (1)

Since only two digits are repeating, we multiply x by 100

100 x = 23.53535... (2)

Subtracting equation (1) from (2), we get,

99 x = 23.3

Example 10: Find an irrational number between 1/7 and 2/7.

Solution:

We know that,

To find an irrational number between 1/7 and 2/7, we find a number which is non-terminating non-recurring lying between them.

Examples are,

0.150150015000150000...,

0.1601011011101111… and so on.

Representing Real Numbers on the Number Line - Process of Successive Magnification

• Represent 2.665 on the number line.

• First divide the portion of line from 2 to 3 in 10 equal parts.

• 2.665 lies between 2.6 and 2.7. Divide this part again into ten equal parts. The first mark will represent 2.61, the next 2.62, and so on.

• Now again the number lies between 2.66 and 2.67. Divide this portion further into 10 equal parts. We can see the fifth divider will be corresponding to 2.665.

Operations on Real Numbers

• The sum or difference of a rational number and an irrational number is irrational.

• The product or quotient of a non-zero rational number with an irrational number is irrational.

• If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.

For any given positive real number x , represent $\sqrt{\mathbit{x}}$ geometrically.

• Mark the distance of x units from a fixed point A on a given line to obtain a point B such that AB = x units.

• From B, mark a distance of 1 unit and mark the new point as C.

• Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC.

• Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = $\sqrt{\mathrm{x}}$.

• Draw an arc with centre B and radius BD, which intersects the number line in E, then BE = $\sqrt{\mathrm{x}}$

Idenities for exponents of real numbers

For positive real numbers a and b,

 3. 4. 5. 6.

Let a > 0 be a real number and p and q be rational numbers. Then

Take examples of

Rationaisation of denominator,

Exponents and powers.

Find rational and irrational numbers between two numbers.

Represent etc.

Visualise $2.\stackrel{̅}{37}$ upto 5 decimal digits etc.