Chapter 1
Number Systems
- Numbers 1, 2, 3…….∞, which are used for counting are called natural numbers. The collection of natural numbers is denoted by N. Therefore, N = {1, 2, 3, 4, 5, ……}.
- When 0 is included with the natural numbers, then the new collection of numbers called is called whole number. The collection of whole numbers is denoted by W. Therefore, W = {0, 1, 2, 3, 4, 5, ……}.
- The negative of natural numbers, 0 and the natural number together constitutes integers. The collection of integers is denoted by I. Therefore, I = {…, -3, -2, -1, 0, 1, 2, 3, ……}.
- The numbers which can be represented in the form of p/q, where q ≠ 0 and p and q are integers are called rational numbers. Rational numbers are denoted by Q. If p and q are co-prime, then the rational number is in its simplest form.
- All-natural numbers, whole numbers and integer are rational number.
- Equivalent rational numbers (or fractions) have same (equal) values when written in the simplest form.
- Rational number between two numbers x and y =x+y2.
- There are infinitely many rational numbers between any two given rational numbers.
- The numbers which are not of the form of p/ q, where q ≠ 0 and p and q are integers are called irrational numbers. For example: √2,√ 7,π, etc.
- Rational and irrational numbers together constitute are called real numbers. The collection of real numbers is denoted by R.
- Irrational number between two numbers x and y
12. Terminating fractions are the fractions which leaves remainder 0 on division.
13. Recurring fractions are the fractions which never leave a remainder 0 on division.
14. The decimal expansion of rational number is either terminating or non-terminating recurring. Also, a number whose decimal expansion is terminating or non-terminating recurring is rational.
15. The decimal expansion of an irrational number is non-terminating non-recurring. Also, a number whose decimal expansion is non-terminating non-recurring is irrational.
16. 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.
17. The process of visualization of numbers on the number line through a magnifying glass is known as the process of successive magnification. This technique is used to represent a real number with non-terminating recurring decimal expansion.
18. Irrational numbers like 2,3,5n, for any positive integer n can be represented on number line by using Pythagoras theorem.
19. If a > 0 is a real number, then a=b means b2 = a and b > 0.
20. For any positive real number x, we have:
21. For every positive real number x, x can be represented by a point on the number line using the following steps:
- Obtain the positive real number, say x.
- Draw a line and mark a point A on it.
- Mark a point B on the line such that AB = x units.
- From B, mark a distance of 1 unit on extended AB and name the new point as C.
- Find the mid-point of AC and name that point as O.
- Draw a semi-circle with centre O and radius OC.
- Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D.
- Length BD is equal to x.
22. Properties of irrational numbers:
- The sum, difference, product and quotient of two irrational numbers need not always be an irrational number.
- Negative of an irrational number is an irrational number.
- Sum of a rational and an irrational number is irrational.
- Product and quotient of a non-zero rational and irrational number is always irrational.
25. The process of removing the radical sign from the denominator of an expression to convert it to an equivalent expression whose denominator is a rational number is called rationalising the denominator.
26. The multiplicating factor used for rationalising the denominator is called the rationalising factor.
27. If a and b are positive real numbers, then
28. The exponent is the number of times the base is multiplied by itself.
29. In the exponential representation am, a is called the base and m is called the exponent or power.
30. Laws of exponents: If a, b are positive real numbers and m, n are rational numbers, then
Numbers
Number: Arithmetical value representing a particular quantity. The various types of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers etc.
Natural Numbers
Natural numbers(N) are positive numbers i.e. 1, 2, 3 ..and so on.
Whole Numbers
Whole numbers (W) are 0, 1, 2, .. and so on. Whole numbers are all Natural Numbers including ‘0’. Whole numbers do not include any fractions, negative numbers or decimals.
Integers
Integers are the numbers that includes whole numbers along with the negative numbers.
Rational Numbers
A number ‘r’ is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.
Irrational Numbers
Any number that cannot be expressed in the form of p/q, where p and q are integers and q≠0, is an irrational number. Examples: √2, 1.010024563…, e, π
Real Numbers
Any number which can be represented on the number line is a Real Number(R). It includes both rational and irrational numbers. Every point on the number line represents a unique real number.
Irrational Numbers
Representation of Irrational numbers on the Number line
Let √x be an irrational number. To represent it on the number line we will follow the following steps:
- Take any point A. Draw a line AB = x units.
- Extend AB to point C such that BC = 1 unit.
- Find out the mid-point of AC and name it ‘O’. With ‘O’ as the centre draw a semi-circle with radius OC.
- Draw a straight line from B which is perpendicular to AC, such that it intersects the semi-circle at point D.
- Length of BD = √x.
Constructions to Find the root of x.
With BD as the radius and origin as the centre, cut the positive side of the number line to get √x.
Identities for Irrational Numbers
Arithmetic operations between:
- rational and irrational will give an irrational number.
- irrational and irrational will give a rational or irrational number.
Example: 2 × √3 = 2√3 i.e. irrational. √3 × √3 = 3 which is rational.
Identities for irrational numbers
Rationalisation is converting an irrational number into a rational number. Suppose if we have to rationalise 1/√a.
Decimal Representation of Rational Numbers
Decimal expansion of Rational and Irrational Numbers
The decimal expansion of a rational number is either terminating or non- terminating and recurring.
Example: 1/2 = 0.5 , 1/3 = 3.33…….
The decimal expansion of an irrational number is non terminating and non-recurring.
Examples: √2 = 1.41421356..
Expressing Decimals as rational numbers
Case 1 – Terminating Decimals
Example – 0.625
Let x = 0.625
If the number of digits after the decimal point is y, then multiply and divide the number by 10y.
So, x = 0.625 × 1000/1000 = 625/1000 Then, reduce the obtained fraction to its simplest form.
Hence, x = 5/8
Case 2: Recurring Decimals
If the number is non-terminating and recurring, then we will follow the following steps to convert it into a rational number:
Example = 1.042
Step 1. Let x = 1.042 ……..(1)
Step 2. Multiply the first equation with 10y, where y is the number of digits that are recurring.
Thus, 100x = 104.242 ………(2)
Steps 3. Subtract equation 1 from equation 2. On subtracting equation 1 from 2, we get99x = 103.2x = 103.2/99 = 1032/990
Which is the required rational number.
Reduce the obtained rational number to its simplest form Thus,
X = 172/165
Also Read: – Class 9th Science : Physics Chapter 5: Sound
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