Expanded Form Of 243546747 A Comprehensive Guide With Examples

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In mathematics, understanding the expanded form of a number is crucial for grasping the concept of place value. Place value refers to the value of a digit based on its position in a number. The expanded form breaks down a number into the sum of each digit's value, making it easier to comprehend the magnitude of each digit and how they contribute to the overall number. This article delves into the expanded form of the number 24,35,46,747, providing a detailed explanation and enhancing your understanding of place value.

The expanded form of a number is a way of writing it to show the value of each digit. This representation helps in understanding the contribution of each digit based on its position in the number. For the number 24,35,46,747, we need to break it down digit by digit, considering the place value of each. The place values, from right to left, are ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, and ten crores. In the Indian numbering system, which is commonly used in the Indian subcontinent, the places are grouped in twos after hundreds, leading to lakhs and crores. The number 24,35,46,747 can be broken down as follows:

  • 2 is in the ten crores place (20,00,00,000)
  • 4 is in the crores place (4,00,00,000)
  • 3 is in the ten lakhs place (30,00,000)
  • 5 is in the lakhs place (5,00,000)
  • 4 is in the ten thousands place (40,000)
  • 6 is in the thousands place (6,000)
  • 7 is in the hundreds place (700)
  • 4 is in the tens place (40)
  • 7 is in the ones place (7)

Therefore, the expanded form of 24,35,46,747 is: 20,00,00,000 + 4,00,00,000 + 30,00,000 + 5,00,000 + 40,000 + 6,000 + 700 + 40 + 7. This expanded representation clarifies the value each digit holds within the number, making it easier to grasp its magnitude and structure. Understanding expanded form is not just an academic exercise; it is crucial for performing arithmetic operations, comparing numbers, and understanding the base-10 system. By recognizing the place value of each digit, we can better perform calculations, estimate values, and appreciate the structure of numbers in our daily lives. In addition, mastering the expanded form is a stepping stone to understanding more complex mathematical concepts, such as scientific notation and algebraic expressions. For example, when dealing with very large numbers in scientific notation, understanding the place value helps in converting between standard and scientific forms. Similarly, in algebra, understanding how numbers are composed of different place values is essential for manipulating and simplifying expressions. Ultimately, the concept of expanded form is a fundamental building block in mathematics that enhances numerical literacy and problem-solving skills. Whether you are a student learning basic arithmetic or a professional dealing with complex calculations, a solid grasp of expanded form will undoubtedly be beneficial.

Breaking Down 24,35,46,747 into Its Place Values

To truly understand the expanded form, we must meticulously break down the number 24,35,46,747 according to its place values. The Indian numbering system, which is used here, groups digits differently than the international system. After the hundreds place, digits are grouped in pairs, leading to the places of lakhs and crores. This grouping affects how we express the expanded form, making it crucial to understand the value of each position. Let's examine each digit in 24,35,46,747:

  • 2 (Ten Crores Place): The digit 2 is in the ten crores place, which means it represents 20 crores or 200,000,000. This is the highest place value in this number, and it carries the most significant weight. Understanding that this single digit contributes 200 million to the total value gives a clear perspective on the scale of the number.
  • 4 (Crores Place): The digit 4 is in the crores place, representing 4 crores or 40,000,000. Adding this to the ten crores place, we already have 24 crores, demonstrating how place values incrementally build the total value of the number.
  • 3 (Ten Lakhs Place): Moving to the lakhs category, the digit 3 is in the ten lakhs place, representing 30 lakhs or 3,000,000. This contribution, while smaller than the crores, is still significant and crucial to the overall value.
  • 5 (Lakhs Place): The digit 5 is in the lakhs place, representing 5 lakhs or 500,000. Combining this with the ten lakhs, we get 35 lakhs, further refining our understanding of the number's composition.
  • 4 (Ten Thousands Place): Shifting focus to the thousands, the digit 4 in the ten thousands place represents 40,000. This is a considerable value and highlights the importance of the thousands category in shaping the number.
  • 6 (Thousands Place): The digit 6 is in the thousands place, representing 6,000. Together with the ten thousands, we have 46,000, continuing to build the number piece by piece.
  • 7 (Hundreds Place): Moving to the hundreds, the digit 7 represents 700. This is the start of the smaller place values but still plays a role in the overall composition of the number.
  • 4 (Tens Place): The digit 4 in the tens place represents 40. This contributes to the final tens value and is essential for precise calculations.
  • 7 (Ones Place): Finally, the digit 7 in the ones place represents 7. This is the smallest place value and completes the number.

By meticulously breaking down the number 24,35,46,747, we uncover the contribution of each digit to the whole. This process not only reinforces the concept of place value but also provides a tangible understanding of how large numbers are constructed. Understanding each digit's value is crucial for comparing numbers, performing arithmetic operations, and appreciating the scale of numerical values in various contexts. This detailed breakdown is fundamental for anyone looking to enhance their mathematical skills and gain a deeper appreciation for numerical structures.

Expressing 24,35,46,747 in Expanded Form

The expanded form of a number is its representation as the sum of each digit multiplied by its corresponding place value. This method is particularly useful for understanding the magnitude of each digit and how it contributes to the overall value of the number. For 24,35,46,747, expressing it in expanded form involves breaking down the number based on the place value of each digit and then writing it as a sum. Let's illustrate this process:

First, we identify the place value of each digit in the number:

  • 2 is in the ten crores place (20,00,00,000)
  • 4 is in the crores place (4,00,00,000)
  • 3 is in the ten lakhs place (30,00,000)
  • 5 is in the lakhs place (5,00,000)
  • 4 is in the ten thousands place (40,000)
  • 6 is in the thousands place (6,000)
  • 7 is in the hundreds place (700)
  • 4 is in the tens place (40)
  • 7 is in the ones place (7)

Now, we express the number as the sum of these values:

24,35,46,747 = (2 × 10,00,00,000) + (4 × 1,00,00,000) + (3 × 10,00,000) + (5 × 1,00,000) + (4 × 10,000) + (6 × 1,000) + (7 × 100) + (4 × 10) + (7 × 1)

Simplifying this, we get:

24,35,46,747 = 20,00,00,000 + 4,00,00,000 + 30,00,000 + 5,00,000 + 40,000 + 6,000 + 700 + 40 + 7

This is the expanded form of 24,35,46,747. The expanded form clearly shows how each digit contributes to the number's total value. It is a valuable tool for reinforcing place value concepts and understanding the structure of large numbers. By writing the number in this form, we can easily see the significance of each position and how it relates to the whole. For example, the '2' in the ten crores place contributes 200 million to the number, while the '7' in the ones place contributes only 7. This clear distinction is what makes the expanded form so useful for teaching and learning about place value. In addition to educational purposes, the expanded form is also helpful in various mathematical operations. For instance, it can simplify the process of adding and subtracting large numbers by breaking them down into their constituent parts. Furthermore, understanding the expanded form is essential for grasping more advanced mathematical concepts, such as scientific notation and polynomial representation. Ultimately, mastering the expanded form is a crucial step in building a strong foundation in mathematics and enhancing numerical literacy. It provides a clear and structured way to understand numbers and their values, making complex mathematical operations more accessible.

The Importance of Expanded Form in Mathematics

The expanded form is not just a method for representing numbers; it is a fundamental concept in mathematics that underpins our understanding of place value and number systems. Understanding the importance of expanded form is crucial for students and anyone looking to deepen their mathematical knowledge. This representation plays a significant role in various mathematical operations and concepts, making it an essential tool for numerical literacy.

Firstly, the expanded form clarifies the value of each digit in a number. By breaking down a number into the sum of its place values, we can see how each digit contributes to the whole. This is particularly important when dealing with large numbers. For instance, in the number 24,35,46,747, the expanded form (20,00,00,000 + 4,00,00,000 + 30,00,000 + 5,00,000 + 40,000 + 6,000 + 700 + 40 + 7) makes it clear that the digit '2' in the ten crores place is worth 200 million, while the digit '7' in the ones place is worth only 7. This understanding is crucial for comparing numbers and performing arithmetic operations.

Secondly, the expanded form is essential for performing arithmetic operations such as addition, subtraction, multiplication, and division. When adding large numbers, for example, understanding place value allows us to align the digits correctly and add them column by column. Similarly, when subtracting, borrowing becomes easier to understand when we recognize the value of each digit based on its place. Multiplication and division also benefit from a clear understanding of expanded form, as it helps in breaking down the numbers into smaller, manageable parts. For example, multiplying 24,35,46,747 by 10 is easily understood as shifting each digit one place to the left, which is a direct consequence of understanding place value.

Moreover, the concept of expanded form is a stepping stone to understanding more advanced mathematical concepts. In algebra, understanding how numbers can be broken down into their place values is crucial for manipulating algebraic expressions. For instance, the number 24,35,46,747 can be seen as a polynomial in base 10: (2 × 10^8) + (4 × 10^7) + (3 × 10^6) + (5 × 10^5) + (4 × 10^4) + (6 × 10^3) + (7 × 10^2) + (4 × 10^1) + (7 × 10^0). This representation is fundamental for understanding polynomial functions and their properties.

Furthermore, the expanded form is vital in understanding different number systems, such as the binary (base-2) and hexadecimal (base-16) systems used in computer science. In these systems, the concept of place value remains the same, but the base changes. Understanding the expanded form in base 10 makes it easier to grasp how numbers are represented and manipulated in other bases.

In conclusion, the importance of expanded form in mathematics cannot be overstated. It is a foundational concept that clarifies place value, simplifies arithmetic operations, and paves the way for understanding more advanced mathematical ideas. Whether you are a student learning basic arithmetic or a professional working with complex calculations, a strong grasp of expanded form will undoubtedly enhance your mathematical skills and understanding.

Conclusion

In summary, the expanded form of 24,35,46,747 is 20,00,00,000 + 4,00,00,000 + 30,00,000 + 5,00,000 + 40,000 + 6,000 + 700 + 40 + 7. This representation vividly illustrates the value of each digit based on its place within the number. Understanding the expanded form is not merely an academic exercise; it's a cornerstone of mathematical comprehension. It empowers us to grasp the structure and magnitude of numbers, making arithmetic operations more intuitive and accessible. This knowledge extends beyond basic calculations, forming a crucial foundation for more advanced mathematical concepts. The expanded form provides a clear and structured method for understanding numbers and their values, making complex mathematical operations more manageable. Whether you're a student mastering arithmetic or a professional dealing with intricate calculations, a solid grasp of expanded form significantly enhances your mathematical skills and overall numerical literacy.