Write The Number Given Its Expanded Notation:${ 8 \cdot 1000 + 4 \cdot 10 + 9 \cdot 1 + 2 \cdot \frac{1}{10} + 3 \cdot \frac{1}{100} + 5 \cdot \frac{1}{1000} + 6 \cdot \frac{1}{10,000} }$

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Introduction

Expanded notation is a way of representing numbers in a more detailed and explicit manner. It involves breaking down a number into its constituent parts, each of which is multiplied by a power of 10. This notation is particularly useful in mathematics, as it allows for easier manipulation and calculation of numbers. In this article, we will explore the concept of expanded notation and how to write a number given its expanded notation.

Understanding Expanded Notation

Expanded notation is a way of representing a number as a sum of its decimal digits, each multiplied by a power of 10. For example, the number 1234 can be represented in expanded notation as:

1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1

This notation is useful because it allows us to easily perform arithmetic operations on numbers. For instance, if we want to add 1234 and 5678, we can simply add the corresponding digits in each number.

Expanded Notation and Its Applications

Expanded notation has numerous applications in mathematics, particularly in algebra and arithmetic. It is used to represent numbers in a more detailed and explicit manner, making it easier to perform calculations and manipulate numbers. Some of the key applications of expanded notation include:

  • Arithmetic operations: Expanded notation makes it easier to perform arithmetic operations such as addition, subtraction, multiplication, and division.
  • Algebraic manipulations: Expanded notation allows us to easily manipulate algebraic expressions and equations.
  • Number theory: Expanded notation is used to represent numbers in number theory, which is a branch of mathematics that deals with the properties and behavior of numbers.

Writing a Number Given Its Expanded Notation

Now that we have a good understanding of expanded notation, let's explore how to write a number given its expanded notation. The expanded notation of a number is represented as a sum of its decimal digits, each multiplied by a power of 10. For example, the expanded notation of the number 1234 is:

1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1

To write a number given its expanded notation, we simply need to evaluate the expression and add up the results. For example, if we want to write the number given its expanded notation:

8 × 1000 + 4 × 10 + 9 × 1 + 2 × 1/10 + 3 × 1/100 + 5 × 1/1000 + 6 × 1/10,000

We can simply evaluate the expression and add up the results:

8 × 1000 = 8000 4 × 10 = 40 9 × 1 = 9 2 × 1/10 = 0.2 3 × 1/100 = 0.03 5 × 1/1000 = 0.005 6 × 1/10,000 = 0.0006

Adding up the results, we get:

8000 + 40 + 9 + 0.2 + 0.03 + 0.005 + 0.0006 = 8040.2356

Therefore, the number given its expanded notation is 8040.2356.

Conclusion

Expanded notation is a powerful tool in mathematics that allows us to represent numbers in a more detailed and explicit manner. It has numerous applications in algebra and arithmetic, and is used to represent numbers in number theory. In this article, we explored how to write a number given its expanded notation, and provided a step-by-step guide on how to evaluate the expression and add up the results. We hope that this article has provided a useful introduction to expanded notation and its significance in mathematics.

Expanded Notation Formula

The expanded notation of a number can be represented as:

a × 10^n + b × 10^(n-1) + c × 10^(n-2) + ... + z × 10^0

where a, b, c, ..., z are the decimal digits of the number, and n is the number of decimal places.

Expanded Notation Example

Let's consider an example of expanded notation:

1234 = 1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1

In this example, the number 1234 is represented as a sum of its decimal digits, each multiplied by a power of 10.

Expanded Notation Practice

Now that we have a good understanding of expanded notation, let's practice writing a number given its expanded notation. Try evaluating the following expressions and adding up the results:

  • 9 × 1000 + 5 × 10 + 2 × 1 + 3 × 1/10 + 4 × 1/100
  • 7 × 1000 + 3 × 100 + 9 × 10 + 2 × 1
  • 1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1 + 5 × 1/10

Frequently Asked Questions

Expanded notation is a powerful tool in mathematics that allows us to represent numbers in a more detailed and explicit manner. However, it can be a bit confusing at first, especially for those who are new to the concept. In this article, we will answer some of the most frequently asked questions about expanded notation.

Q: What is expanded notation?

A: Expanded notation is a way of representing numbers in a more detailed and explicit manner. It involves breaking down a number into its constituent parts, each of which is multiplied by a power of 10.

Q: How do I write a number given its expanded notation?

A: To write a number given its expanded notation, you simply need to evaluate the expression and add up the results. For example, if you want to write the number given its expanded notation:

8 × 1000 + 4 × 10 + 9 × 1 + 2 × 1/10 + 3 × 1/100 + 5 × 1/1000 + 6 × 1/10,000

You can simply evaluate the expression and add up the results:

8 × 1000 = 8000 4 × 10 = 40 9 × 1 = 9 2 × 1/10 = 0.2 3 × 1/100 = 0.03 5 × 1/1000 = 0.005 6 × 1/10,000 = 0.0006

Adding up the results, you get:

8000 + 40 + 9 + 0.2 + 0.03 + 0.005 + 0.0006 = 8040.2356

Therefore, the number given its expanded notation is 8040.2356.

Q: What are the benefits of using expanded notation?

A: The benefits of using expanded notation include:

  • Easier arithmetic operations: Expanded notation makes it easier to perform arithmetic operations such as addition, subtraction, multiplication, and division.
  • Simpler algebraic manipulations: Expanded notation allows us to easily manipulate algebraic expressions and equations.
  • Improved understanding of numbers: Expanded notation helps us to understand numbers in a more detailed and explicit manner.

Q: How do I convert a number from expanded notation to standard notation?

A: To convert a number from expanded notation to standard notation, you simply need to evaluate the expression and add up the results. For example, if you want to convert the number given its expanded notation:

8 × 1000 + 4 × 10 + 9 × 1 + 2 × 1/10 + 3 × 1/100 + 5 × 1/1000 + 6 × 1/10,000

You can simply evaluate the expression and add up the results:

8 × 1000 = 8000 4 × 10 = 40 9 × 1 = 9 2 × 1/10 = 0.2 3 × 1/100 = 0.03 5 × 1/1000 = 0.005 6 × 1/10,000 = 0.0006

Adding up the results, you get:

8000 + 40 + 9 + 0.2 + 0.03 + 0.005 + 0.0006 = 8040.2356

Therefore, the number given its expanded notation is 8040.2356.

Q: Can I use expanded notation with fractions?

A: Yes, you can use expanded notation with fractions. For example, if you want to represent the number 3.14 in expanded notation, you can write it as:

3 × 1 + 1 × 1/10 + 4 × 1/100

This represents the number 3.14 in expanded notation.

Q: How do I use expanded notation in real-life situations?

A: Expanded notation is used in a variety of real-life situations, including:

  • Finance: Expanded notation is used to represent financial numbers, such as interest rates and investment returns.
  • Science: Expanded notation is used to represent scientific numbers, such as measurements and calculations.
  • Engineering: Expanded notation is used to represent engineering numbers, such as dimensions and calculations.

Conclusion

Expanded notation is a powerful tool in mathematics that allows us to represent numbers in a more detailed and explicit manner. It has numerous benefits, including easier arithmetic operations, simpler algebraic manipulations, and improved understanding of numbers. In this article, we have answered some of the most frequently asked questions about expanded notation, including how to write a number given its expanded notation, the benefits of using expanded notation, and how to convert a number from expanded notation to standard notation. We hope that this article has provided a useful introduction to expanded notation and its applications.