Write The Following Numbers In Expanded Form: (a) 42350009 (b) 6627319 (c) 425255

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Mathematics, the language of the universe, often presents numbers in various forms. One such form is the expanded form, which provides a detailed breakdown of a number based on the place value of its digits. Understanding the expanded form is crucial for grasping the structure of numbers and performing arithmetic operations with ease. In this comprehensive guide, we will delve into the concept of expanded form and illustrate it with examples, specifically addressing the given questions:

(a) 4,23,50,009 (b) 66,27,319 (c) 4,25,255

Decoding Expanded Form: A Foundation

The expanded form of a number is a way of writing it to show the value of each digit. It breaks down the number into the sum of each digit multiplied by its place value. To understand this better, let's first define place value.

Place Value: In our decimal number system, each digit in a number has a place value that corresponds to its position. Starting from the rightmost digit, the place values are ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, crores, and so on. Each place value is ten times greater than the place value to its right. This system is fundamental to how we represent and understand numerical quantities.

For example, in the number 123, the digit 3 is in the ones place, the digit 2 is in the tens place, and the digit 1 is in the hundreds place. Thus, the value of 3 is 3 × 1 = 3, the value of 2 is 2 × 10 = 20, and the value of 1 is 1 × 100 = 100. The expanded form of 123 is therefore 100 + 20 + 3.

Why is Expanded Form Important?

The expanded form is not just a mathematical exercise; it has several practical applications:

  1. Understanding Number Structure: It helps in visualizing the composition of a number and the significance of each digit.
  2. Simplifying Arithmetic: It can make addition, subtraction, and other arithmetic operations easier to understand and perform, especially with large numbers.
  3. Foundation for Algebra: It lays the groundwork for understanding algebraic expressions and equations, where numbers are represented by variables.
  4. Enhancing Number Sense: It improves overall number sense and the ability to work with numbers confidently.

Applying Expanded Form: Solving the Given Examples

Now that we have a solid understanding of expanded form, let's apply it to the given numbers.

(a) 4,23,50,009

The number 4,23,50,009 is a large number in the Indian numbering system. To write it in expanded form, we break it down digit by digit, noting the place value of each digit.

  • 4 is in the crores place (10,000,000), so its value is 4 × 10,000,000 = 40,000,000.
  • 2 is in the ten-lakhs place (1,000,000), so its value is 2 × 1,000,000 = 2,000,000.
  • 3 is in the lakhs place (100,000), so its value is 3 × 100,000 = 300,000.
  • 5 is in the ten-thousands place (10,000), so its value is 5 × 10,000 = 50,000.
  • 0 is in the thousands place (1,000), so its value is 0 × 1,000 = 0.
  • 0 is in the hundreds place (100), so its value is 0 × 100 = 0.
  • 0 is in the tens place (10), so its value is 0 × 10 = 0.
  • 9 is in the ones place (1), so its value is 9 × 1 = 9.

Thus, the expanded form of 4,23,50,009 is:

40,000,000 + 2,000,000 + 300,000 + 50,000 + 0 + 0 + 0 + 9

This detailed expansion clearly shows how each digit contributes to the overall value of the number. The zeroes act as placeholders, maintaining the correct magnitude of the other digits.

(b) 66,27,319

The number 66,27,319 is another example that allows us to practice expanded form. We proceed similarly, identifying the place value of each digit.

  • 6 is in the ten-lakhs place (1,000,000), so its value is 6 × 1,000,000 = 6,000,000.
  • 6 is in the lakhs place (100,000), so its value is 6 × 100,000 = 600,000.
  • 2 is in the ten-thousands place (10,000), so its value is 2 × 10,000 = 20,000.
  • 7 is in the thousands place (1,000), so its value is 7 × 1,000 = 7,000.
  • 3 is in the hundreds place (100), so its value is 3 × 100 = 300.
  • 1 is in the tens place (10), so its value is 1 × 10 = 10.
  • 9 is in the ones place (1), so its value is 9 × 1 = 9.

Therefore, the expanded form of 66,27,319 is:

6,000,000 + 600,000 + 20,000 + 7,000 + 300 + 10 + 9

This example further reinforces the concept of expanded form, demonstrating how each digit's position dictates its contribution to the total number.

(c) 4,25,255

Lastly, let's consider the number 4,25,255. This example includes repeated digits, which highlights the importance of considering the place value independently for each digit.

  • 4 is in the lakhs place (100,000), so its value is 4 × 100,000 = 400,000.
  • 2 is in the ten-thousands place (10,000), so its value is 2 × 10,000 = 20,000.
  • 5 is in the thousands place (1,000), so its value is 5 × 1,000 = 5,000.
  • 2 is in the hundreds place (100), so its value is 2 × 100 = 200.
  • 5 is in the tens place (10), so its value is 5 × 10 = 50.
  • 5 is in the ones place (1), so its value is 5 × 1 = 5.

Thus, the expanded form of 4,25,255 is:

400,000 + 20,000 + 5,000 + 200 + 50 + 5

This final example underscores the consistency of the expanded form method, regardless of the digits involved. Each digit's value is determined solely by its place in the number.

Conclusion: Mastering Expanded Form

In conclusion, the expanded form is a powerful tool for understanding the structure and value of numbers. By breaking down numbers into the sum of their place values, we gain a deeper appreciation for how our number system works. The examples provided—4,23,50,009, 66,27,319, and 4,25,255—illustrate the method in action, covering a range of number sizes and digit patterns. Mastering expanded form not only enhances mathematical skills but also builds a solid foundation for more advanced concepts in algebra and beyond. Always remember to consider each digit's place value to accurately represent a number in its expanded form. This understanding will serve you well in your mathematical journey.

By understanding and practicing with expanded form, students and learners can build a stronger foundation in mathematics. This method is not just about writing numbers differently; it's about understanding the inherent structure and value within each number, making arithmetic and other mathematical operations more intuitive and manageable.