• TS 6^th Maths Concept
• TS 7^th Maths Concept
• TS 8^th Maths Concept
• TS 9^th Maths Concept
• TS 10^th Maths Concept

Studying maths in IX class successfully means that children take responsibility for their own learning and learn to apply the concepts to solve problems.

This note is designed by the ‘Basics in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear.

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1.REAL NUMBERS

Rational numbers:-

• The numbers which are written in the form of, where p, q are integers and q≠ 0 are called rational numbers. Rational numbers are denoted by Q.

ex:-  3/2, 3/5, 2, 1 and so on

• Natural numbers, Whole numbers, and Integers are rational numbers.

• The rational numbers do not have a unique representation.

  Representation of rational number:

      Represent   

To find a rational number between given numbers:

 Mean method:- A rational number between two numbers a and b is

  Ex:- insert two rational number between 1 and 2

To find a rational number in a single step:-

 Ex:- insert two rational number between 1 and 2

 To find two rational numbers, we 1 and 2 as rational numbers with same denominator 3     (∵ 1 + 2 = 3)

 

The decimal form of rational numbers:

• Note:- Every rational number can be expressed as a terminating decimal or non-terminating repeating decimal.
• Converting decimal form into a fraction:

1. Terminating decimals:-  (i) 1.2 = 12/10 = 6/5

                  (ii) 1.35 =135/100 = 135/100 = 27/20

2. Non-Terminating repeating decimals:-

Irrational numbers:

• The numbers which are not written in the form of, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q^I or S.

• Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:-

   Calculation of square roots:

• There is a reference of irrationals in the calculation of square roots in Sulbha Sutra.
• Procedure to find   value:

Representing irrational numbers on a number line:

          Ex:- Locate    on a number line

• At ‘O’ draw a unit square OABC on a number line with each side 1 unit in length.

• By Pythagoras theorem                          OB² = OA² + AB²

=  1² + 1²

OB² = 2

OB =

• Using a compass with centre O and radius OB, draw an arc on the right side to O intersecting the number line at the point
• The location of is now at k.

• Note:-  If a and b are two positive rational numbers such that ab is not a perfect square, this an irrational number lies between a and b.   

Real numbers

• The collection of all rational and irrational numbers is called real numbers.
• Real numbers cover all the points on the number line.
• Every real number is represented by a unique point on the number line.
• Ex:-   are some examples of real numbers.

Representing real numbers on the number line through successive magnifications:-

locating 2. 775 on a number line

Operation on real numbers

• The sum, difference, product and quotient of irrational numbers need not be an irrational number.
• Irrational numbers are not closed under addition, subtraction, multiplication, and division.
• For any two real numbers a and b

Rationalizing the denominator:

• Rationalizing factor(R.F):-If the product of two irrational numbers is rational, then each of the two is the rationalizing factor to others.

• The rationalizing factor of a given irrational number is not unique. It is convenient to use the simplest of all R.F.s of given irrational number.
• Note:-

Law of exponents for real numbers:

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