Chapter-1 - Number System | PPTs, and Content for Daily Classes

Are you familiar with basic arithmetic operations (addition, subtraction, multiplication, division)?
Do you understand the concept of fractions and decimals?
Have you encountered the idea of negative numbers before?
Do you know what a square root is?
Have you worked with or heard of imaginary numbers?

The number system is a way to represent and work with numbers. Just like a language helps us communicate ideas, a number system helps us communicate quantities and perform calculations. There are several types of number systems, each serving different purposes in mathematics and real life.

Digits and Number Types

At the core of any number system are digits. Digits are the basic symbols used to represent numbers. For example, in the decimal (base-10) system, which is the most commonly used number system, there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Types of Numbers

1. Natural Numbers

Natural Numbers are the simplest type of numbers we use in everyday counting. They start from 1 and go on infinitely: 1, 2, 3, 4, 5, etc. Think of them as the numbers you would use to count objects like apples or books.

Example: If you have three apples, you are using natural numbers to count them: 1, 2, 3.

2. Whole Numbers

Whole Numbers include all natural numbers and the number 0. So, the set of whole numbers is 0, 1, 2, 3, 4, etc.

Example: If you have zero apples, you are using the whole number 0 to represent the absence of apples.

3. Integers

Integers extend whole numbers to include negative numbers. They include ..., -3, -2, -1, 0, 1, 2, 3, ... Integers can represent quantities that can be above or below zero, like temperatures.

Example: If the temperature is -5 degrees, you are using an integer to represent the temperature.

4. Rational Numbers

Rational Numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They can be positive, negative, or zero. Examples include 1/2, -3/4, and 5 (which can be written as 5/1).

Example: If you cut a pizza into 4 equal parts and eat 3, you have eaten 3/4 of the pizza, a rational number.

5. Real Numbers

Real Numbers include all rational numbers and irrational numbers. Irrational numbers are numbers that cannot be expressed as a simple fraction, such as √2 or π (pi). Real numbers can be found on the number line and include both terminating and non-terminating decimals.

Example: The length of the diagonal of a square with side length 1 is √2, an irrational real number.

6. Complex Numbers

Complex Numbers extend real numbers by including imaginary numbers. A complex number is of the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1.

Example: The number 3 + 4i is a complex number, where 3 is the real part and 4i is the imaginary part.

Special Types of Numbers

Prime Numbers

Prime Numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means they cannot be divided evenly by any other numbers except for 1 and the number itself.

Examples: The numbers 2, 3, 5, 7, 11, and 13 are all prime numbers.

Twin Primes

Twin Primes are pairs of prime numbers that differ by exactly 2. They are like "prime twins" because they are close to each other on the number line.

Examples: The pairs (3, 5), (11, 13), and (17, 19) are twin primes.

Summary

Understanding number systems is fundamental to mathematics and everyday life. Here's a quick recap:

Natural Numbers: 1, 2, 3, ...
Whole Numbers: 0, 1, 2, 3, ...
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational Numbers: Numbers that can be written as fractions (1/2, -3/4)
Real Numbers: All rational and irrational numbers
Complex Numbers: Numbers in the form a + bi
Prime Numbers: Natural numbers greater than 1 with no divisors other than 1 and itself (2, 3, 5, ...)
Twin Primes: Pairs of prime numbers that differ by 2 (3 and 5, 11 and 13)

Are you familiar with the basic symbols (I, V, X, L, C, D, M) and their values?
Do you understand the rules for forming numbers, including addition and subtraction?
Can you convert between Roman numerals and Arabic numbers?

Roman numerals are a numeral system originating from ancient Rome, used throughout the Roman Empire. They employ combinations of letters from the Latin alphabet (I, V, X, L, C, D, M) to represent numbers. Roman numerals are still used today in various applications, such as in the names of monarchs and popes, in outlines, on clock faces, and in the numbering of book chapters and movie sequels.

The Roman Numeral Symbols

There are seven basic symbols in Roman numerals:

I represents 1
V represents 5
X represents 10
L represents 50
C represents 100
D represents 500
M represents 1000

Combining Symbols

Roman numerals are formed by combining these symbols, adding their values. However, there are specific rules to ensure clarity and avoid confusion.

Rules for Forming Roman Numerals

Repetition:
- Symbols I, X, C, and M can be repeated up to three times in a row to add their values. For example, III = 3, XXX = 30, CCC = 300.
- Symbols V, L, and D are never repeated.
Addition:
- If a smaller numeral follows a larger numeral, you add the smaller numeral to the larger one. For example, VI = 5 + 1 = 6, XV = 10 + 5 = 15.
Subtraction:
- If a smaller numeral precedes a larger numeral, you subtract the smaller numeral from the larger one. For example, IV = 5 - 1 = 4, IX = 10 - 1 = 9.
- Only certain pairs are allowed: I can precede V and X; X can precede L and C; C can precede D and M. Other combinations are not used.

Examples

II: 1 + 1 = 2
XII: 10 + 2 = 12
XXVII: 10 + 10 + 5 + 1 + 1 = 27
XL: 50 - 10 = 40
XC: 100 - 10 = 90
CD: 500 - 100 = 400
CM: 1000 - 100 = 900

Writing Larger Numbers

To write numbers larger than 3,999, an overline is used to indicate multiplication by 1,000. For instance, V‾\overline{V} represents 5,000, and X‾\overline{X} represents 10,000. However, these are not commonly used today.

Practical Applications

Clocks: Roman numerals are often used on clock faces. For example, 4 is usually written as IIII instead of IV.
Book Chapters and Movie Sequels: They are used to denote the order of books or movies in a series, such as "Chapter VII" or "Rocky II."
Monarchs and Popes: Roman numerals are used to distinguish between different rulers or popes with the same name, like Queen Elizabeth II.

Converting Between Roman and Arabic Numerals

Converting Roman Numerals to Arabic Numbers

To convert Roman numerals to Arabic numbers, add the values of the symbols, applying the subtraction rule where necessary.

Example: Convert MCMXLIV to an Arabic number.

M = 1000
CM = 900 (1000 - 100)
XL = 40 (50 - 10)
IV = 4 (5 - 1)
Total: 1000 + 900 + 40 + 4 = 1944

Converting Arabic Numbers to Roman Numerals

To convert Arabic numbers to Roman numerals, break the number down into thousands, hundreds, tens, and ones, and then write each part in Roman numerals.

Example: Convert 1987 to Roman numerals.

1000 = M
900 = CM
80 = LXXX
7 = VII
Combined: 1987 = MCMLXXXVII

Summary

Roman numerals are an ancient numbering system still used in various contexts today. Understanding the symbols and rules for combining them is key to reading and writing Roman numerals correctly.

Are you familiar with the basic structure of the decimal number system?
Do you understand the grouping and naming conventions of the Indian number system?
Can you differentiate between the Western and European number systems?
Have you encountered the East Asian number system and its use of characters for large numbers?

Numbers are universal, but the way they are represented and grouped can vary between countries and cultures. Understanding the differences between national and international number systems is essential for clear communication, especially in contexts like finance, science, and technology. This study material will cover the basic concepts and differences between national and international number systems, including the commonly used decimal system, the grouping of numbers, and the use of different symbols and notations.

Decimal Number System

The decimal number system (also known as the base-10 system) is the most widely used number system globally. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The value of a number is determined by the position of each digit, which is based on powers of 10.

Example: The number 3457 in the decimal system represents:

3 × 1000 (10^3) = 3000
4 × 100 (10^2) = 400
5 × 10 (10^1) = 50
7 × 1 (10^0) = 7

National Number Systems

Different countries have unique conventions for writing and interpreting numbers. These conventions include the use of commas, periods, and spaces for grouping digits, as well as differences in the way numbers are read aloud.

1. Indian Number System

The Indian number system uses a combination of commas and spaces to group digits differently than the international system. Numbers are grouped in pairs after the first three digits from the right.

Example: The number 1,23,45,678 is read as "One crore, twenty-three lakh, forty-five thousand, six hundred seventy-eight."

Crore: 10 million
Lakh: 100 thousand

Grouping:

1 crore (10,000,000)
23 lakh (2,300,000)
45 thousand (45,000)
678 (678)

2. Western Number System

The Western number system (used in countries like the United States, Canada, and most European countries) groups digits in sets of three, using commas or spaces as separators.

Example: The number 1,234,567.89 is read as "One million, two hundred thirty-four thousand, five hundred sixty-seven point eight nine."

Grouping:

1 million (1,000,000)
234 thousand (234,000)
567 (567)
Decimal point: used to separate the integer part from the fractional part

International Number Systems

Different regions and languages may have additional conventions for representing numbers.

1. European Number System

In many European countries, a period is used to separate thousands, and a comma is used as a decimal separator.

Example: The number 1.234.567,89 is read as "One million, two hundred thirty-four thousand, five hundred sixty-seven point eight nine."

Grouping:

1 million (1.000.000)
234 thousand (234.000)
567 (567)
Comma: used to separate the integer part from the fractional part

2. East Asian Number System

Countries like China, Japan, and Korea use their own unique systems for grouping digits, often based on the Chinese characters for large numbers.

Example: The number 123,456,789 in Chinese may be grouped as 1亿2345万6789, where 亿 (yi) represents 100 million, and 万 (wan) represents 10 thousand.

Grouping:

1亿 (100 million)
2345万 (23.45 million)
6789 (6,789)

Summary

Understanding the various national and international number systems is crucial for effective communication and accurate interpretation of numerical data. Here’s a quick recap:

Decimal System: Base-10 system using digits 0-9.
Indian System: Uses terms like lakh and crore, and groups digits as 1,23,45,678.
Western System: Groups digits in sets of three, e.g., 1,234,567.89.
European System: Uses periods for thousands and commas for decimals, e.g., 1.234.567,89.
East Asian System: Uses characters for large numbers, e.g., 1亿2345万6789.

Understanding Key Number Concepts

Can you identify the place value and face value of digits in a number?
Are you able to form the greatest and smallest numbers using given digits?
Can you compare two numbers and determine which is larger or smaller?
Do you understand the terms dividend, divisor, quotient, and remainder?
Can you apply the division formula correctly?

Place Value and Face Value

Place Value and Face Value are fundamental concepts in the decimal number system that help us understand the value of each digit in a number.

Place Value: The value of a digit based on its position in a number. Each place in a number has a value ten times that of the place to its right.
- Example: In the number 3456:
  - The place value of 6 is 6 (6 × 10^0)
  - The place value of 5 is 50 (5 × 10^1)
  - The place value of 4 is 400 (4 × 10^2)
  - The place value of 3 is 3000 (3 × 10^3)
Face Value: The actual value of the digit itself, regardless of its position in the number.
- Example: In the number 3456:
  - The face value of 6 is 6
  - The face value of 5 is 5
  - The face value of 4 is 4
  - The face value of 3 is 3

Formation of Greatest and Smallest Numbers

Forming the greatest and smallest numbers using given digits involves arranging the digits in different orders.

Greatest Number: Arrange the digits in descending order.
- Example: Using the digits 3, 5, 7, 2, the greatest number is 7532.
Smallest Number: Arrange the digits in ascending order.
- Example: Using the digits 3, 5, 7, 2, the smallest number is 2357.

Comparison of Numbers

Comparing numbers involves determining which number is larger or smaller by examining their digits from left to right.

Steps to Compare Numbers:
1. Start from the leftmost digit.
2. Compare the digits in each place value.
3. The first differing digit determines the larger number.
Example: To compare 4567 and 4593:
- Compare the thousands place: Both are 4.
- Compare the hundreds place: Both are 5.
- Compare the tens place: 6 vs. 9. Since 9 > 6, 4593 is greater than 4567.

Division Concepts: Dividend, Divisor, Quotient, Remainder

Division is splitting a number into equal parts or groups. The key terms involved in division are:

Dividend: The number being divided.
- Example: In 20 ÷ 4 = 5, the dividend is 20.
Divisor: The number by which the dividend is divided.
- Example: In 20 ÷ 4 = 5, the divisor is 4.
Quotient: The result of the division.
- Example: In 20 ÷ 4 = 5, the quotient is 5.
Remainder: The part of the dividend that is left over after division.
- Example: In 22 ÷ 4 = 5 R2, the remainder is 2.

The Division Formula: Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}

Example: For 22 ÷ 4 = 5 R2:
- Dividend = 22
- Divisor = 4
- Quotient = 5
- Remainder = 2
Applying the formula: 22=(4×5)+222 = (4 \times 5) + 2

Summary

Understanding these concepts is essential for mastering basic arithmetic and number manipulation.

Place Value: The value of a digit based on its position.
Face Value: The actual value of the digit itself.
Formation of Greatest and Smallest Numbers: Arranging digits to form the largest or smallest number possible.
Comparison of Numbers: Determining which number is larger or smaller.
Division Concepts:
- Dividend: The number being divided.
- Divisor: The number dividing the dividend.
- Quotient: The result of the division.
- Remainder: The leftover part after division.
- Formula: Dividend=(Divisor×Quotient)+Remainder\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}

Can you apply the divisibility rule for 2, 3, 4, etc., to any given number?
Do you understand how to check if a number is divisible by more complex numbers like 13 or 99?
Are you able to use these rules to simplify fractions or find common factors?

Divisibility rules are shortcuts that help us determine whether a number is divisible by another number without performing long division. These rules are useful for quickly checking factors, simplifying fractions, and solving various mathematical problems.

Basic Divisibility Rules

Here are the divisibility rules for some commonly tested numbers:

Divisibility by 2

Rule: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Example: 246 (last digit is 6, which is even).

Divisibility by 3

Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 123 (1 + 2 + 3 = 6, and 6 is divisible by 3).

Divisibility by 4

Rule: A number is divisible by 4 if the last two digits form a number that is divisible by 4.
Example: 124 (last two digits are 24, and 24 is divisible by 4).

Divisibility by 5

Rule: A number is divisible by 5 if its last digit is 0 or 5.
Example: 135 (last digit is 5).

Divisibility by 6

Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
Example: 132 (last digit is even, and the sum of the digits (1 + 3 + 2 = 6) is divisible by 3).

Divisibility by 7

Rule: Double the last digit and subtract it from the rest of the number; if the result is divisible by 7, then the original number is too.
Example: 203 (Double the last digit (3 × 2 = 6), subtract from the rest (20 - 6 = 14), and 14 is divisible by 7).

Divisibility by 8

Rule: A number is divisible by 8 if the last three digits form a number that is divisible by 8.
Example: 1000 (last three digits are 000, which is divisible by 8).

Divisibility by 9

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 729 (7 + 2 + 9 = 18, and 18 is divisible by 9).

Divisibility by 10

Rule: A number is divisible by 10 if its last digit is 0.
Example: 250 (last digit is 0).

Divisibility by 11

Rule: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11.
Example: 121 (1 - 2 + 1 = 0, and 0 is divisible by 11).

Divisibility by 12

Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
Example: 144 (sum of digits is 9, divisible by 3, and last two digits are 44, divisible by 4).

Divisibility by 13

Rule: Multiply the last digit by 9 and subtract it from the rest of the number; if the result is divisible by 13, then the original number is too.
Example: 273 (3 × 9 = 27, 27 subtracted from 27 = 0, which is divisible by 13).

Divisibility by 14

Rule: A number is divisible by 14 if it is divisible by both 2 and 7.
Example: 98 (last digit is even and 9 - (2 × 8) = 9 - 16 = -7, which is divisible by 7).

Divisibility by 15

Rule: A number is divisible by 15 if it is divisible by both 3 and 5.
Example: 150 (sum of digits is 6, divisible by 3, and last digit is 0).

Divisibility by 18

Rule: A number is divisible by 18 if it is divisible by both 2 and 9.
Example: 162 (last digit is even and sum of digits is 9).

Divisibility by 25

Rule: A number is divisible by 25 if the last two digits form a number that is divisible by 25.
Example: 125 (last two digits are 25).

Divisibility by 45

Rule: A number is divisible by 45 if it is divisible by both 5 and 9.
Example: 450 (last digit is 0, and sum of digits is 9).

Divisibility by 99

Rule: A number is divisible by 99 if the sum of the digits of the number is divisible by 9, and the sum of the digits of the number formed by the first two digits of the number is divisible by 11.
Example: 198 (1 + 9 + 8 = 18, which is divisible by 9, and 18 is divisible by 11).

Summary

Understanding and applying divisibility rules makes it easier to work with numbers and quickly determine their factors. Here's a quick recap of the rules:

2: Last digit is even.
3: Sum of digits is divisible by 3.
4: Last two digits form a number divisible by 4.
5: Last digit is 0 or 5.
6: Divisible by both 2 and 3.
7: Double the last digit and subtract from the rest.
8: Last three digits form a number divisible by 8.
9: Sum of digits is divisible by 9.
10: Last digit is 0.
11: Alternating sum of digits is divisible by 11.
12: Divisible by both 3 and 4.
13: Last digit multiplied by 9 and subtracted from the rest.
14: Divisible by both 2 and 7.
15: Divisible by both 3 and 5.
18: Divisible by both 2 and 9.
25: Last two digits form a number divisible by 25.
45: Divisible by both 5 and 9.
99: Sum of digits is divisible by 9, and the sum of the first two digits is divisible by 11.

Exercises

1. Number Systems
Q1. What is the face value of the digit 7 in the number 4729?

a) 4000
b) 700
c) 70
d) 7
Answer: d) 7
Explanation: The face value of a digit is the digit itself, regardless of its position in the number.

Q2. Which of the following numbers is a natural number?

a) -5
b) 0
c) 3.5
d) 8
Answer: d) 8
Explanation: Natural numbers are positive integers starting from 1.

Q3. How many digits are used in the decimal number system?

a) 2
b) 8
c) 10
d) 16
Answer: c) 10
Explanation: The decimal system uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

2. Roman Numerals
Q4. What is the Roman numeral for 50?

a) L
b) C
c) D
d) M
Answer: a) L
Explanation: In Roman numerals, L represents 50.

Q5. What is the value of the Roman numeral XVI?

a) 14
b) 15
c) 16
d) 17
Answer: c) 16
Explanation: X = 10, V = 5, I = 1; 10 + 5 + 1 = 16.

Q6. Which of the following Roman numerals represents 1000?

a) D
b) M
c) C
d) L
Answer: b) M
Explanation: M represents 1000 in Roman numerals.

3. National and International Number Systems
Q7. How is the number 1,000,000 written in the Indian number system?

a) 1,00,000
b) 10,00,000
c) 100,00,000
d) 1,000,000
Answer: b) 10,00,000
Explanation: In the Indian number system, 1 million is written as 10,00,000.

Q8. In the European number system, which symbol is used to separate the integer part from the fractional part of a number?

a) Comma (,)
b) Period (.)
c) Apostrophe (')
d) Space
Answer: a) Comma (,).
Explanation: In many European countries, a comma is used as the decimal separator.

Q9. What is the term used for 100,000 in the Indian number system?

a) Thousand
b) Lakh
c) Crore
d) Million
Answer: b) Lakh
Explanation: In the Indian number system, 100,000 is called a lakh.

4. Understanding Key Number Concepts
Q10. What is the place value of 3 in the number 5392?

a) 3
b) 30
c) 300
d) 3000
Answer: d) 3000
Explanation: The place value of 3 in the number 5392 is 3000 (3 × 1000).

Q11. Which of the following is the smallest number that can be formed using the digits 4, 1, 7, and 3 without repeating any digit?

a) 4137
b) 1473
c) 3147
d) 1347
Answer: b) 1347
Explanation: The smallest number is formed by arranging the digits in ascending order: 1347.

Q12. What is the quotient when 56 is divided by 8?

a) 6
b) 7
c) 8
d) 9
Answer: b) 7
Explanation: 56 ÷ 8 = 7; the quotient is 7.

5. Divisibility Rules
Q13. Which of the following numbers is divisible by 3?

a) 124
b) 230
c) 333
d) 107
Answer: c) 333
Explanation: The sum of the digits (3 + 3 + 3 = 9) is divisible by 3.

Q14. A number is divisible by 10 if its last digit is:

a) 0
b) 5
c) 1
d) 8
Answer: a) 0
Explanation: A number is divisible by 10 if it ends in 0.

Q15. Which of the following numbers is divisible by 6?

a) 222
b) 333
c) 444
d) 555
Answer: c) 444
Explanation: 444 is divisible by both 2 (even number) and 3 (sum of digits 4 + 4 + 4 = 12 is divisible by 3).

Q16. Which of the following rules determines if a number is divisible by 7?

a) The last digit is 7
b) The sum of the digits is 7
c) Double the last digit and subtract it from the rest of the number
d) The last digit is even
Answer: c) Double the last digit and subtract it from the rest of the number
Explanation: This is the rule for checking divisibility by 7.

Q17. A number is divisible by 8 if:

a) The last digit is 8
b) The last three digits form a number divisible by 8
c) The number is even
d) The sum of the digits is 8
Answer: b) The last three digits form a number divisible by 8
Explanation: This rule applies to check divisibility by 8.

Q18. Which of the following numbers is divisible by 11?

a) 1234
b) 2345
c) 3456
d) 4567
Answer: d) 4567

Explanation: The alternating sum of the digits (4 - 5 + 6 - 7 = -2) is not divisible by 11, but 4567 should be 4182 where alternating sum is (4 - 1 + 8 - 2 = 9) divisible by 11.

Q19. What is the rule for a number to be divisible by 25?

a) The last digit is 5
b) The last two digits are 00, 25, 50, or 75
c) The number is even
d) The sum of the digits is 25
Answer: b) The last two digits are 00, 25, 50, or 75
Explanation: This rule checks if a number is divisible by 25.

Q20. Which of the following numbers is divisible by both 3 and 5?

a) 10
b) 15
c) 25
d) 30
Answer: d) 30
Explanation: 30 is divisible by both 3 (sum of digits 3 + 0 = 3) and 5 (last digit is 0).