NUMBER SYSTEM

Number System

Decimal System: Base 10 system that uses 10 digits (0-9) to represent numbers. Each digit's position signifies its weight or value in powers of 10.
Binary System: Base 2 system using only two digits, 0 and 1. It's widely used in computers where each digit represents a power of 2.
Octal System: Base 8 system using digits 0-7. Each digit's position represents a power of 8.
Hexadecimal System: Base 16 system using digits 0-9 and letters A-F to represent values 10-15. Used in computing for representing large binary numbers in a more compact form.

Types of Numbers:

Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, ...).
Whole Numbers (W): Natural numbers including zero (0, 1, 2, 3, ...).
Integers (Z): Whole numbers and their negatives, including zero (-3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers (Q): Numbers that can be expressed as a fraction (p/q), where p and q are integers and q is not zero.
Irrational Numbers (Not in Q): Numbers that can't be expressed as a fraction and have non-repeating/non-terminating decimal expansions (π, √2).
Real Numbers (R): All rational and irrational numbers, including integers, fractions, decimals, and square roots.
Complex Numbers (C): Numbers in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√(-1))

Operations on Numbers

Exponents and Powers

Exponents and powers are significant concepts within the number system, providing a shorthand way to represent repeated multiplication.

Exponents: An exponent indicates how many times a number (the base) is multiplied by itself. It's denoted as a superscript to the right of the base number. For instance:

$a n$ represents the base $a$ raised to the power of $n$ .
Here, $a$ is the base and $n$ is the exponent or power.

Basic Laws and Operations:

Product of Powers: When multiplying two powers with the same base, you can add their exponents.

Example: $a m \times a n = a(^{m + n)}$

Quotient of Powers: When dividing two powers with the same base, you can subtract their exponents.

Example:

Power of a Power: To raise a power to another power, multiply the exponents.

Example:

(a^{m})^{n} = a^{m \times n}

Power of a Product: When a product is raised to an exponent, each factor is raised to that exponent.

Example:

(ab)^{n} = a^{n} \times b^{n}

Power of a Quotient: When a quotient is raised to an exponent, the numerator and denominator are each raised to that exponent.

Example:

(a/b)^{n} = a ^{n /b n}

Application in Numbers:

Negative Exponents: An exponent can be negative, indicating a reciprocal of the base raised to the positive exponent.

Example:

a^{- n} = 1/a n

Fractional Exponents: Exponents can be fractions or roots, indicating roots of the base.

Example:

Scientific Notation: Expressing very large or very small numbers in a convenient format using powers of 10.

Example:

(5.6 multiplied by

, which is

LCM and HCF

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest multiple that is divisible by each of these numbers without leaving a remainder. It's used to find a common denominator in fractions and to solve various mathematical problems involving multiple numbers.

There are a few methods to find the LCM:

Method 1: Prime Factorization

Prime Factorization: Express each number in its prime factorization form.
LCM Calculation: Take all the distinct prime factors with their highest powers to find the LCM.

Method 2: Using Multiplication

List Multiples: Write down multiples of each number until you find a common multiple.
Choose the Least: Find the smallest common multiple among those multiples.

Example: Finding LCM of 8 and 12 using Multiplication

Multiples of 8: 8, 16, 24, 32, 40, ...
Multiples of 12: 12, 24, 36, 48, ...

The least common multiple of 8 and 12 is 24.

LCM can also be calculated using the formula involving the Greatest Common Divisor (GCD) and the relationship between LCM and GCD for two numbers, $a$ and $b$ :

This formula shows that the product of two numbers divided by their GCD results in their LCM.

Finding the LCM is important in various mathematical calculations, especially in arithmetic, algebra, and fraction operations. It's used in simplifying fractions, adding and subtracting fractions with different denominators, and solving equations involving multiple variables or quantities

Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest number that divides each of them without leaving a remainder. It's the highest number that is a factor of both numbers.

There are several methods to find the HCF:

Method 1: Prime Factorization

Prime Factorization: Express each number in its prime factorization form.
Common Factors: Identify the common prime factors and their lowest powers.
HCF Calculation: Multiply these common prime factors.

Example: Finding HCF of 36 and 48 using Prime Factorization

$2^{2} \times 3^{2}$ (Prime factors: 2 and 3)
$2^{4} \times 3^{1}$ (Prime factors: 2 and 3)

Identify common factors with the lowest powers:

Common factors: 2 (lowest power = 2^2) and 3 (lowest power = 3^1)
HCF

Method 2: Using Division

Division Method: Divide the larger number by the smaller one.
Remainder: Find the remainder.
Repeat Process: Divide the divisor by the remainder.
Continue Division: Repeat until the remainder becomes zero.
HCF Calculation: The divisor at the point where the remainder becomes zero is the HCF.

Example: Finding HCF of 36 and 48 using Division

48÷36 = $1$ with a remainder of

$36 \div 12 = 3 with a remainder of$

Since the remainder became zero, the divisor at this point ( $12$ ) is the HCF of 36 and 48.

Formula: HCF can also be calculated using the prime factorization of the numbers and the relationship between HCF and LCM for two numbers, $a$ and $b$ :

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

This formula demonstrates that the product of HCF and LCM of two numbers is equal to the product of those numbers. Therefore, if you know the LCM, you can calculate the HCF and vice versa

The rule for finding out HCF and LCM of fractions

The rules for finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of fractions involve determining the HCF and LCM of their numerators and denominators separately.

Finding HCF of Fractions:

When finding the HCF of fractions, consider the numerators and denominators separately.

HCF of Numerators: Find the HCF of the numerators.
HCF of Denominators: Find the HCF of the denominators.

The HCF of the fractions will be the fraction formed by the HCF of the numerators divided by the HCF of the denominators.

For example: Let's say we have fractions a/b and c/d

Finding LCM of Fractions:

To find the LCM of fractions:

LCM of Numerators: Find the LCM of the numerators.
LCM of Denominators: Find the LCM of the denominators.

The LCM of the fractions will be the fraction formed by the LCM of the numerators divided by the HCF of the denominators.

For example: Considering fractions a/b and c/d

"The concept of a number line is pivotal in elementary numerical understanding. It represents a straight line extending from negative infinity on the left to positive infinity on the right.

Key Mathematical Representations:

Even numbers: Expressed as 2n.
Odd numbers: Defined by the formula 2n-1.
Multiples of 4: Represented as 4n, and so forth.

Prime Factors: Utilized to express numbers in a standardized form.

Standard or Canonical Form: For instance, determining the standard form of 144.

General Notation for Two or Three-Digit Numbers: In mathematical equations, the representation of two or three-digit numbers is crucial. For instance, a two-digit number 'AB' is expressed as 10A + B, with 'A' in the tens place, signifying the value of 'AB'. Similarly, a three-digit number 'ABC' is represented as 100A + 10B + C, depicting its value.

The BODMAS Rule: This rule governs the sequence of mathematical operations – Brackets, followed by Order (Division and Multiplication), and finally Addition and Subtraction."

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that helps find the remainder when a polynomial is divided by a linear factor. It establishes a relationship between the value of a polynomial at a certain point and the remainder obtained when dividing the polynomial by a linear expression of the form $x - a$ .

Statement of the Remainder Theorem:

Let $f (x)$ be a polynomial of degree $n$ with coefficients $a_{0}, a_{1}, a_{2}, \dots, a_{n}$ and $a$ be any real number. When $f (x)$ is divided by $x - a$ , the remainder obtained is $f (a)$ .

Mathematical Representation:

Given a polynomial $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ of degree $n$ , if it is divided by $x - a$ , the remainder $R$ is equal to $f (a)$ .

Mathematically, the relationship can be represented as:

$f (x) = (x - a) \times Q (x) + R$

Where:

$Q (x)$ is the quotient obtained after dividing $f (x)$ by $x - a$ .
$R$ is the remainder.
$R = f (a)$

Application:

The Remainder Theorem is primarily used to find the remainder when a polynomial is divided by a linear expression without performing the full polynomial division.

Example: Let's consider a polynomial $x^{3} - 4 x^{2} + 5 x - 2$ . Using the Remainder Theorem, find the remainder when $f (x)$ is divided by $x - 3$ .

Solution:

Evaluate $f (3)$ to find the remainder.
$f (3) = (3)^{3} - 4 (3)^{2} + 5 (3) - 2$
$f (3) = 27 - 36 + 15 - 2 = 4$

Therefore, when $f (x)$ is divided by $x - 3$ , the remainder is $4$ .

Practice Questions on Number System

Question 1:

What is the smallest natural number that is divisible by each of the numbers from 1 to 10?

A) 210
B) 220
C) 240
D) 250

Question 2:

2.If $x$ is a positive integer such that $x^{2} + 10 x + 24$ is a perfect square, what is the value of $x$ ?

A) 2
B) 4
C) 6
D) 8

Question 3:

If the product of two consecutive positive even integers is 168, what is the smaller integer?

A) 12
B) 14
C) 16
D) 18

Question 4:

If $a - b = 3$ and $a^{2} - b^{2} = 21$ , what is the value of $a + b$ ?

A) 3
B) 6
C) 9
D) 12

Question 5:

If $x$ and $y$ are positive integers such that $x^{2} - 3 x y + 2 y^{2} = 0$ , what is the value of x/y?

A) 1
B) 2
C) 3
D) 4

Answers:

A) 210
B) 4
B) 14
C) 9
A) 1.

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NUMBER SYSTEM

NUMBER SYSTEM

Operations on Numbers

LCM and HCF

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