Revision Notes on Real Numbers
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Decimal Expration of Rational Number (2)
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Rational Numbers on the basis of sign (2)
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Properties of Rational Numbers (4)
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Additive Inverse योग प्रतिलोम– If sum of two rational number is zero than they are called additive inverse.
Ex- 14 + (-14) = 0,
Note- To gate additive inverse of any no change the sign.
Multiplicative Inverse गुणज प्रतिलोम– If product of two rational number is one than they are called multiplicative inverse.
Ex- 14 × = 1,
Note- Reciprocal of number is multiplicative inverse of itself.
Euclid’s Division Lemma
It is basically the restatement of the usual division system. The formal statement for this is-
For each pair of given positive integers a and b, there exist unique whole numbers q and r which satisfies the relation
a = bq + r, 0 ≤ r < b, where q and r can also be Zero.
where ‘a’ is a dividend, ‘b’ is divisor, ‘q’ is quotient and ‘r’ is remainder.
∴ Dividend = (Divisor x Quotient) + Remainder
Real Numbers- The combination of rational and irrational numbers in number system is called natural numbers.
* It is denoted by R.
Natural Numbers- The counting numbers are called natural numbers.
i.e. 5, 6, 7, 8, ……….
* It is denoted by N.
Whole numbers- All non-negative counting numbers including zero are called whole numbers.
i.e. 0, 1, 2, 3, 4, 5, …………….
* It is denoted by W.
Integers- All negative and non-negative numbers including zero altogether known as integers.
i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
* It is denoted by Z.
* The sum, product, and difference of any two integers is also an integer. But this is not true for division just try 1 ÷ 2.
* The sum their additive inverse is zero. Ex {1+ (-1) = 0, 12 + (-12) = 0}
Rational numbers परिमेय संख्या – Those numbers which can be written in the form where p and q are integers but q≠ 0 are called rational numbers.
* Every integer p is also a rational number, can be written as
* It is denoted by Q.
Irrational numbers अपरिमेय संख्या – Those numbers which cannot be written in the form where p and q are integers but q≠ 0 are called irrational numbers.
Ex- √2, √3 etc.
* The decimal expansion of a rational number is either terminating or non-terminating recurring.
* If p is a prime number and p divides a2 , then p is one of the prime factors of a2 which divides a, where a is a positive integer.
* If p is a positive number and not a perfect square, then √n is definitely an irrational number.
* If p is a prime number, then √p is also an irrational number.
* The decimal form of irrational numbers is non-terminating and non-repeating.
Algorithm कलन विधि- An algorithm gives us some definite steps to solve a particular type of problem in a well-defined manner.
Lemma प्रमेयिका- A lemma is a statement which is already proved and is used for proving other statements.
Euclid’s Division Algorithm
This concept is based on Euclid’s division lemma. This is the technique to calculate the HCF (Highest common factor) of given two positive integers m and n,
To calculate the HCF of two positive integers’ m and n with m > n, the following steps are followed:
Step 1: Apply Euclid’s division lemma to find q and r where m = nq + r, 0 ≤ r < n.
Step 2: If the remainder i.e. r = 0, then the HCF will be ‘n’ but if r ≠ 0 then we have to apply Euclid’s division lemma to n and r.
Step 3: Continue with this process until we get the remainder as zero. Now the divisor at this stage will be HCF(m, n). Also, HCF (m, n) = HCF (n, r), where HCF (m, n) means HCF of m and n.
The Fundamental Theorem of Arithmetic
We can factorize each composite number as a product of some prime numbers and of course, this prime factorization of a natural number is unique as the order of the prime factors doesn’t matter.
* HCF of given numbers is the highest common factor among all which is also known as GCD i.e. greatest common divisor.
* LCM of given numbers is their least common multiple.
* If we have two positive integers ‘m’ and ‘n’ then the property of their HCF and LCM will be:
* HCF (m, n) × LCM (m, n) = m × n.
Rational Number and their Decimal Expansions
* Let y be a real number whose decimal expansion terminates into a rational number which we can express in the form of a/b, where a and b are coprime, and the prime factorization of the denominator b has the powers of 2 or 5 or both like 2n5m, where n, m are non-negative integers.
* Let y be a rational number in the form of y = a/b, so that the prime factorization of the denominator b is of the form 2n5m, where n, m are non-negative integers then y has a terminating decimal expansion.
* Let y = a/b be a rational number, if the prime factorization of the denominator b is not in the form of 2n2m, where n, m are non-negative integers then y has a non-terminating repeating decimal expansion.
* The decimal expansion of every rational number is either terminating or a non-terminating repeating.
* The decimal form of irrational numbers is non-terminating and non-repeating.
Terminating decimals शांत दशमल:- The rational numbers with a finite decimal part are called finite or terminating decimals.
Ex- 5/2 = 2.5
Non-Terminating decimals अशांत दशमल:- The rational numbers with an infinite decimal part are called infinite or non-terminating decimals.
Ex-
NOTE-
* The decimal expansion of an irrational number is non terminating non-recurring.
* If n is a natural number other than a perfect square, then 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.
- Obtain all positive real numbers (say).
- Draw a line and mark a point P on it.
- Make a point Q on the line such that PQ = units.
- From point Q marka distance of 1 unit and mark the new point as R.
- Find the mid-point of PR and mark the point as O.
- Draw a circle with centre O and radius OR.
- Draw a line perpendicular to PR passing through Q and intersecting the semi-circle at S. Length QS is equal to