Real Numbers: Important Formulas

1. Natural Numbers: These are counting numbers. N = {1,2,3,4,5...}

2. Whole numbers: Counting Numbers + Zero. W = {0,1,2,3,4,5...}

3. Integers: Negative and Positive Numbers. Z = {... -5, -4, -3, -2, -1, 0,1,2,3,4,5...}

4. Positive Integers: Z+ = {0,1,2,3,4,5...}

5. Negative Integers: Z- = {... -5, -4, -3, -2, -1}

6. Rational Numbers: A number is said to be rational number(Q) if it can be expressed in the form of $\frac{p}{q}$ where p and q are integers and q ≠ 0.

e.g. $\frac{1}{2},\frac{-3}{7}, 0, 7 $ etc.

8. Irrational Numbers: A number is said to be irrational number if it cannot be expressed in the form of $\frac{p}{q}$

e.g. π, $\sqrt{2}...$

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

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

❍ Euclid’s division algorithm is a technique to compute the Highest Common Factor $(HCF)$ of two given positive integers.

❍ To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

- Step 1:
- Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.
- Step 2:
- If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
- Step 3:
- Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

$HCF(a, b)$ = Product of the smallest power of each common prime factor in the numbers.

$LCM(a, b)$ = Product of the greatest power of each prime factor, involved in the numbers.

If $HCF(a,b) = 1$, then a and b are co-primes.

$HCF( a , b ) \times LCM ( a , b ) = a \times b$

Every composite number can be expressed $(factorized)$ as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

● Let p be a prime number. If p divides a² , then p divides a, where a is a positive integer.

● $\sqrt{2},\sqrt{3},\sqrt{5} $ are irrational

● Let x be a rational number whose decimal expansion terminates. Then we can express x in the form pq, where p and q are coprime, and the prime factorization of q is of the form 2^{n} 5^{m}, where n, m are non-negative integers

● Let x = pq be a rational number, such that the prime factorization of q is of the form 2^{n} 5^{m}, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

● Let x = pq be a rational number, such that the prime factorization of q is not of the form 2^{n} 5^{m}, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating $(recurring)$.