Click The Below Image
To Attempt The Online Test
👇👇👇👇👇
Review Of Concepts
Introduction:
In the realm of forming numbers or numerals, we employ the digits 0 through 9. These particular digits are commonly referred to as "ones." The numerals crafted using the digits 1 through 9 are recognized as Hindu Arabic numbers.
Numbers play a pivotal role in quantifying the various aspects of our daily lives. The numerical system employed in our nation is none other than the Hindu-Arabic numerals. It's fascinating to note that Aryabhatta is credited with pioneering the place value concept of Hindu Arabic numerals, while Brahmagupta introduced the pivotal symbol for zero. These numerals, fundamentally developed in India, found acceptance in Arabian cultures as well, thus earning them the moniker "Hindu Arabic numerals."
Place Value:
The significance of a digit within a number, determined by its position, is termed its place value. Here's a breakdown:
- Ten Thousands
- Thousands
- Hundreds
- Tens
- Ones
The place value of a digit in a given number is the product of the digit and its respective place. When a digit shifts one place to the left, its value escalates tenfold. For instance, the value of the rightmost digit is "one," while the adjacent place to its left represents "ten," equivalent to 10 times the value of one. Similarly, the place to the left of the tens place is "hundred," which signifies 10 times the value of 100, and so forth.
Example:
To illustrate, let's determine the place value of the digits within the number 2847:
- The place value of 7 in 2847 is 7 ones.
- The place value of 4 is 4 tens, amounting to 40.
- The place value of 8 is 8 hundreds, totaling 800.
- The place value of 2 is 2 thousands, which is 2000.
Face Value:
Every digit has a face value equivalent to the digit itself. For instance, the face value of 4 in 45 is 4, and in 896, the face value of 8 is 8.
Example:
Let's calculate the place value and face value of the digits in 7608:
- The place value of 8 is 8 ones, equal to 8.
- The face value of 8 is 8.
- The place value of 0 is 0 tens.
- The face value of 0 is 0.
- The place value of 6 is 6 hundreds. Its face value is 6.
- The place value of 7 is 7 thousands, and its face value is 7.
Expanded Form and Standard (Short) Form:
Numbers can be expressed in two forms: expanded and standard. The expanded form represents a number as the sum of its place values, while the standard form combines the face values of each digit in the correct order.
Example:
Short Form: 4628 Expanded Form: 4000 + 600 + 20 + 8
Example:
Now, let's write the following numbers in their expanded forms: a) 6573 b) 3104 c) 8635 d) 9735
Solution:
a) 6573 = 6000 + 500 + 70 + 3 b) 8635 = 8000 + 600 + 30 + 5 c) 3104 = 3000 + 100 + 4 d) 9735 = 9000 + 700 + 30 + 5
Example:
Convert the expression "9000 + 200 + 40 + 2" into standard form.
Solution:
9000 + 200 + 40 + 2 = 9242
Number Names:
The name of a number is derived from the arrangement of its digits. For instance, the number name of 87654 is "Eighty-seven thousand, six hundred and fifty-four."
Number Number Names:
- 10: Ten
- 20: Twenty
- 50: Fifty
- 100: Hundred
- 500: Five Hundred
- 1000: One Thousand
- 5000: Five Thousand
- 10,000: Ten Thousand
Skip Counting:
Skip counting involves counting forwards or backward by a number other than 1, facilitating quicker counting and aiding in learning multiplication tables.
(a) Skip counting of 2: 2, 4, 6, 8, 10, ... (b) Skip counting of 3: 3, 6, 9, 12, 15, ... (c) Skip counting in fives starting from 50: 55, 60, 65, 70, ... (each number is five more than the previous one)
Skip Count Backward:
Counting in 3s: 30, 27, 24, 21, 18, ...
Skip Count Forward:
Counting in 4s: 100, 104, 108, 112, ...
Successor:
The successor of a number is obtained by adding 1 to that number. For instance, the successor of 1 is 1 + 1 = 2.
Predecessor:
Conversely, the predecessor of a number is obtained by subtracting 1 from that number. For example, the predecessor of 4 is 4 - 1 = 3.
Comparing Numbers:
When comparing numbers, we consider the number of digits. If two numbers have different numbers of digits, the one with more digits is greater.
Example:
Comparing 796 and 97, we find that 796 is greater than 97, which can be denoted as 796 > 97.
If two numbers have the same number of digits, we start comparing from left to right, looking at the first digit, then the second digit, and so on.
Example:
Comparing 6363 and 4129, we determine that 6363 > 4129.
Ordering of Numbers:
When arranging numbers, we can follow the same rules for comparison. Arranging numbers from smallest to largest is known as ascending order, while arranging them from largest to smallest is called descending order.
Example:
Arrange the following numbers in ascending and descending order: 1205, 787, 6085, 1348.
Solution:
Ascending Order: 787 < 1205 < 1348 < 6085 Descending Order: 6085 > 1348 > 1205 > 787
Ascending and Descending Numbers:
Arranging numbers in ascending order means organizing them from the smallest to the largest, while arranging numbers in descending order means arranging them from the largest to the smallest.
Example:
Given the series of numbers: 30, 12, 18, 17, 22, 48, 40, and 28.
Ascending order: 12, 17, 18, 22, 28, 30, 40, 48 Descending order: 48, 40, 30, 28, 22, 18, 17, 12
Rounding Off Numbers:
Rounding off involves approximating numbers to a certain place value.
Rounding to the nearest 10:
- 68 is rounded to 70 (as the ones digit is greater than or equal to 5)
- 152 is rounded to 150 (as the ones digit is less than 5)
Rounding to the nearest 100:
- 678 is rounded to 700 (as the tens digit is greater than or equal to 5)
- 819 is rounded to 800 (as the tens digit is less than 5)
Rounding to the nearest 1000:
- 2913 is rounded to 3000 (as the hundreds digit is greater than or equal to 5)
- 3215 is rounded to 3000 (as the hundreds digit is less than 5)
Number Sense (4-Digit Numbers):
In the realm of numbers, we are acquainted with ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits, commonly known as "ones," serve as building blocks for numbers.
The maximum 3-digit number is 999, which can be represented as 9 hundreds + 9 tens + 9 ones. By adding 1 to 999, we attain the smallest 4-digit number, which is 1000. Thus, 1000 is the successor of 999.
Constructing Numbers with Given Digits: Greatest and Smallest:
To create the greatest 4-digit number, arrange the digits in descending order, placing the largest digit in the thousands place, followed by the next largest in the hundreds place, and so on. Conversely, to form the smallest 4-digit number, arrange the digits in ascending order.
Hindu-Arabic or Indian Number System:
The Indian or Hindu-Arabic number system, with a base of 10, is the foundation of our numerical representation. Each digit's position is ten times greater than the preceding one. For instance, in the number 4563, the digit 6 is in the tens place, which is 10 times the value of the ones place (represented by 100). Similarly, the digit 4 is in the hundreds place, equivalent to 10 times the value of 1000.
International Number System:
In the international numbering system, large numbers are divided into periods such as ones, thousands, millions, and so on, for ease of comprehension. A comma is typically used to separate these periods. For instance, we represent numbers like 1 (ones), 10 (tens), 100 (hundreds), 1,000 (thousands), 10,000 (ten thousands), 100,000 (hundred thousands), 1,000,000 (one million), 10,000,000 (ten million), 100,000,000 (hundred million), and so forth.
Types of Numbers:
- Natural Numbers: These are counting numbers, starting from 1, with no upper limit.
- Whole Numbers: Inclusive of 0, these encompass the set of natural numbers starting from 0.
- Prime Numbers: Numbers with only two factors, 1 and themselves, like 2, which is the sole even prime number.
- Even Numbers: Divisible by 2, like 2, 4, 6, 8, etc.
- Odd Numbers: Not divisible by 2, like 1, 3, 5, 7, etc.
- Twin Primes: Prime numbers with a difference of 2, such as 3 and 5, 11 and 13.
- Composite Numbers: Numbers with more than two factors, aside from 1 and themselves.
Regional Numerals:
Numbers can be represented using various symbols and digits, known as numerals. The choice of numerals depends on the regional numbering system. For example, Roman numerals utilize distinct symbols, while Hindu-Arabic numerals are the standard in many cultures.
Roman Numerals:
Roman numerals are a numeral system that originated in ancient Rome and were widely used throughout the Roman Empire. They are still used today in various contexts, such as on clock faces, book chapters, movie titles, and more. Roman numerals are a non-positional numeral system, which means that the value of a numeral depends on its position relative to other numerals. Here are some key aspects of Roman numerals:
Basic Symbols: Roman numerals use a set of basic symbols to represent numbers. These symbols are:
- I: Represents 1
- V: Represents 5
- X: Represents 10
- L: Represents 50
- C: Represents 100
- D: Represents 500
- M: Represents 1000
Formation Rules: Roman numerals are formed by combining these basic symbols according to specific rules:
- Symbols are usually written from left to right in descending order of value.
- However, there are exceptions to this rule for subtractive notation, where a smaller numeral is placed before a larger one to represent subtraction.
Additive vs. Subtractive Notation: Roman numerals can be expressed using additive notation or subtractive notation. Additive notation involves using only the basic symbols to add up the value of a number, while subtractive notation involves using a smaller symbol before a larger one to indicate subtraction. Subtracting is only allowed with certain pairs of numerals.
No Zero: Roman numerals do not have a symbol for zero. This absence of a zero symbol makes arithmetic operations more complex.
Combining Numerals: To represent numbers larger than 1000, Roman numerals combine the basic symbols and their values.
Limitations: Roman numerals are not well-suited for arithmetic operations and complex calculations. They are primarily used for representing numbers in a symbolic and decorative manner.
Modern Usage: Roman numerals are still commonly used in various contexts, such as in the titles and copyright dates of books, movie credits, and on clock faces. For example, Super Bowl events are often identified with Roman numerals.
Educational and Historical Significance: Learning Roman numerals is often part of basic education and is seen as a way to connect with ancient Roman culture and history.
Roman numerals are a unique and historical numeral system that continues to have a place in modern culture and symbolism. They offer a distinctive way of representing numbers, and their usage persists in various fields today.
