Formula Sheet – Real Numbers – Class X

1. Given positive integers a and  b, there exist unique integers  q and r satisfying
a=bq+r,0\le r<b . This result is known as Euclid’s division lemma.

2. An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

3. A lemma is a proven statement used for proving another statement.

4. HCF of two positive integers  a and b is the largest positive integer d that divides both a and b .

5. Euclid’s Division Algorithm : To obtain the HCF of two positive integers, say c and d with  c>d, we follow the steps below :

Step 1. Apply Euclid’s division lemma to find q and r where c=dq+r,0\le r<d .

Step 2. If r=0 , then d is the HCF of c and d . If r\ne0 , then apply Euclid’s division lemma to d and  r.

Step 3. Continue this process till the remainder is zero. The divisor at this stage will be the required HCF.

6. The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. Or the prime factorisation of a natural number is unique, except for the order of its factors.

7. Any number which cannot be expressed in the form \dfrac{p}{q} where  p and  q are integers and p\ne 0 is called an irrational number.

8. Let p be a prime number. If p divides  a, then  p divides a , where  a is a positive integer.

9. The sum or difference of a rational and an irrational number is irrational.

10. The product and quotient of a non-zero rational number and an irrational number is irrational.

11. Let x be a rational number whose decimal expansion terminates. Then  x can be expressed in the form \dfrac{p}{q} where p and  q are coprime and the prime factorisation of  q is of the form 2^n5^m , where  n and m are non negative integers.

12. Let x=\dfrac{p}{q} be a rational number such that the prime factorisation of q is of the form  2^n5^m, where  n and  m are non negative integers. Then,  x has a decimal expansion which terminates.

13. If x=\dfrac{p}{q} is a rational number, such that the prime factorisation of  q is of the form  2^m5^n, where m and n are whole numbers. If m=n , then the decimal expansion of x will terminate after  m places of decimal. If m>n , then the decimal expansion of x will terminate after m places of decimal. If n>m , then the decimal expansion of  x will terminate after n places of decimal.

14. Let x=\dfrac{p}{q} be a rational number, such that the prime factorisation of q is not of the form  2^n5^m, where  n and  m are non negative integers. Then  x has a decimal expansion which is non terminating repeating (recurring).

15. The decimal expansion of every rational number is either terminating or non-terminating repeating.

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