Class 9

Number Systems

1.       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, ……}.

2.       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, ……}.

3.       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, ……}.

4.       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.

5.       All-natural numbers, whole numbers and integer are rational number.

6.       Equivalent rational numbers (or fractions) have same (equal) values when written in the simplest form.

7.       Rational number between two numbers x and y .

8.       There are infinitely many rational numbers between any two given rational numbers.

9.       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:  etc.

10.  Rational and irrational numbers together constitute are called real numbers. The collection of real numbers is denoted by R.

11.  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 , for any positive integer n can be represented on number line by using Pythagoras theorem.

19.  If a > 0 is a real number, then  means b2 = a and b > 0.

20.  For any positive real number x, we have:

21.  For every positive real number x,  can be represented by a point on the number line using the following steps:

                    i.         Obtain the positive real number, say x.

                  ii.         Draw a line and mark a point A on it.

                iii.         Mark a point B on the line such that AB = x units.

               iv.         From B, mark a distance of 1 unit on extended AB and name the new point as C.

                 v.         Find the mid-point of AC and name that point as O.

               vi.         Draw a semi-circle with centre O and radius OC.

             vii.         Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D.

           viii.         Length BD is equal to .