1118 words - 5 pages

Grade 9 Number Systems

Natural numbers The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is denoted by N. N = {1, 2, 3, …} Whole numbers If we include zero to the set of natural numbers, then we get the set of whole numbers. The set of whole numbers is denoted by W. W = {0, 1, 2, …} Integers The collection of numbers … –3, –2, –1, 0, 1, 2, 3 … is called integers. This collection is denoted by Z, or I. Z = {…, –3, –2, –1, 0, 1, 2, 3, …}

Rational numbers Rational numbers are those which can be expressed in the form integers and q Note: 1.

12 12 3 4 , where the HCF of 4 and 5 is 1 15 15 3 5 12 4 and are equivalent rational numbers (or fractions) 15 5 a ...view middle of the document...

So, the number line is also called the real number line. Example: Locate 6 on the number line. Solution: It is seen that:

6 5

2

12

To locate 6 on the number line, we first need to construct a length of 5 .

5 22 1

By Pythagoras Theorem:

OB2 OB OA 2 AB2 5 22 12 4 1 5

Steps: (a) Mark O at 0 and A at 2 on the number line, and then draw AB of unit length 5 perpendicular to OA. Then, by Pythagoras Theorem, OB (b) Construct BD of unit length perpendicular to OB. Thus, by Pythagoras Theorem, OD

5

2

12

6

(c) Using a compass with centre O and radius OD, draw an arc intersecting the number line at point P. Thus, P corresponds to the number 6 .

Real numbers and their decimal expansions: The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating). Moreover, a number whose decimal expansion is terminating or non-terminating repeating is rational. Example:

3 2 15 8 4 3 24 13 1.5 1.875 Terminating Terminating Non – terminating recurring Non-terminating recurring

1.333....... 1.3

1.846153846153 1.846153

Example:

Show that 1.23434 …. can be written in the form and q 0. Solution:

Let x 1.23434..... 1.234 1

p , where p and q are integers q

Here, two digits are repeating. Multiplying (1) by 100, we get: 100x = 123.43434……… =122.2 + 1.23434 …….. Subtracting (1) from (2), we get:

99 x 122.2 x 122.2 99 1222 990 661 495 611 495

(2)

Thus,1.234

The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non- recurring is irrational. Example: 2.645751311064……. is an irrational number Representation of real numbers on the number line Example: Visualize 3.32 on the number line, upto 4 decimal places. Solution:

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