# Number Systems class 9 Notes Mathematics

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## CBSE Guide Number Systems class 9 Notes

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## 9 Mathematics notes Chapter 1 Number Systems

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CBSE Class 09 Mathematics
Revision Notes
CHAPTER – 1
NUMBER SYSTEMS

1 Rational Numbers

2 Irrational Numbers

3 Real Numbers and their Decimal Expansions

4 Operations on Real Numbers

5 Laws of Exponents for Real Numbers

• Natural numbers are : 1, 2, 3, …………….. denoted by N.
• Whole numbers are : 0, 1, 2, 3, ……………… denoted by W.
• Integers : ……. -3, -2, -1, 0, 1, 2, 3, ……………… denoted by Z.
• Rational numbers – All the numbers which can be written in the form ${p \over q}$ are called rational numbers where p and q are integers and $q \ne 0.$ Every integer p is also a rational number, can be written as ${p \over 1}.$
• Irrational numbers – A number is called irrational, if it cannot be written in the form ${p \over q}$  where p and q are integers and $q \ne 0.$
• The decimal expansion of a rational number is either terminating or non terminating recurring. Thus we say that a number whose decimal expansion is either terminating or non terminating recurring is a rational number.
• Terminating decimals: The rational numbers with a finite decimal part or for which the long division terminates after a finite number of steps are known as finite or terminating decimals.
• Non-Terminating decimals: The rational numbers with an infinite decimal part or for which the long division does not terminate even after an infinite number of steps are known as infinite or non-terminating decimals
• The decimal expansion of a irrational number is non terminating non recurring.
• All the rational numbers and irrational numbers taken together make a collection of real numbers.
• A real number is either rational or irrational.
• If r is rational and s is irrational then r+s, r–s, r.s are always irrational numbers but ${r \over s}$ may be rational or irrational.
• If n is a natural number other than a perfect square, then $\sqrt n$ is a irrational number.
• Negative of an irrational number is an irrational number.
• There is a real number corresponding to every point on the number line. Also, corresponding to every real number there is a point on the number line.
• Every irrational number can be represented on a number line using Pythagoras theorem.
• For every positive real number $x,$ $\sqrt x$ can be represented by a point on the number line by using the following steps:
1. Obtain all positive real numbers $x$ (say).
2. Draw a line and mark a point P on it.
3. Make a point Q on the line such that PQ = $x$ units.
4. From point Q marka distance of 1 unit and mark the new point as R.
5. Find the mid-point of PR and mark the point as O.
6. Draw a circle with centre O and radius OR.
7. Draw a line perpendicular to PR passing through Q and intersecting the semi-circle at S. Length QS is equal to $\sqrt x$
• Rationalization means to remove square root from the denominator.

$\frac{{3 + \sqrt5 }}{{\sqrt 2 }}$ to remove we will multiply both numerator & denominator by $\sqrt 2$

${1 \over {a \pm \sqrt b }}$  its rationalization factor $a \mp \sqrt b$