Innings2
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Chapter 1: Real Numbers

    Introduction

    The Fundamental Theorem of Arithmetic

    Irrational Numbers

    Exponentials Revisted

    Exercise 1.1

    Exercise 1.2

    Exercise 1.3

    Exercise 1.4

    Exercise 1.5

    What We Have Discussed

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Chapter 1: Real Numbers > What We Have Discussed

What We Have Discussed

1. Division Algorithm : For any two positive integers a and b, there exist whole numbers q (quotient) and r (remainder) such that: a = bq + r, 0 ≤ r b

This theorem helps in division and finding remainders efficiently.

2. Fundamental Theorem of Arithmetic:

Every composite number can be uniquely expressed as a product of prime numbers, regardless of the order of factors. This principle forms the basis of factorization.

3. Divisibility Rule for Primes

If p is a prime number and divides a2 (where a is a positive integer), then p must also divide .

4. Terminating Decimal Expansions of Rational Numbers

A rational number x can be written as: x = pq

where p and q are coprime. If the denominator q has a prime factorization of the form 2m 5n (where m and n are non-negative integers), then x has a decimal expansion.

5. Non-Terminating, Repeating Decimals

If the denominator q in x = pq has prime factors other than 2 or 5, then its decimal representation is and repeating.

6. Definition of Logarithms

A logarithm is defined as: logax = n if an = x

where a and x are positive numbers, and a 1.

7. Laws of Logarithms

For any positive real numbers a, x, and y (where a ≠ 1), the following logarithmic rules hold:

  1. Product Rule: loga (xy) = loga x + loga y

  2. Quotient Rule: loga xy = loga x - loga y

  3. Power Rule: loga xm = m loga x

  4. Exponential Form: alogaN = N

  5. Log of 1: loga 1 = 0

  6. Log of Base Number: loga a = 1

8. Applications of Logarithms

Logarithms are widely used in engineering, science, business, and economics for calculations involving exponential growth, interest rates, and data analysis.