Expanded Form Of Numbers Explained With Examples

Understanding the expanded form of numbers is a fundamental concept in mathematics. It helps break down numbers into their individual place values, making it easier to grasp the value of each digit. This article will delve into the expanded form of numbers, providing a detailed explanation and examples to solidify your understanding. We will specifically address the expanded forms for the numbers in the table provided, ensuring a comprehensive grasp of the concept.

What is the Expanded Form of a Number?

The expanded form of a number is a way of writing it that shows the value of each digit. It decomposes a number into the sum of its individual place values. Each digit is multiplied by its corresponding place value (ones, tens, hundreds, thousands, etc.), and these products are then added together. This representation clarifies the magnitude and contribution of each digit to the overall value of the number.

For instance, consider the number 345. In expanded form, it would be written as (3 x 100) + (4 x 10) + (5 x 1). This clearly shows that the 3 represents 3 hundreds, the 4 represents 4 tens, and the 5 represents 5 ones. Understanding this concept is crucial for various mathematical operations, including addition, subtraction, multiplication, and division.

The concept of expanded form is not just limited to whole numbers; it can also be applied to decimals. In the case of decimals, the place values extend to the right of the decimal point, representing tenths, hundredths, thousandths, and so on. For example, the number 12.34 can be written in expanded form as (1 x 10) + (2 x 1) + (3 x 0.1) + (4 x 0.01). This representation is particularly useful when dealing with decimal operations and understanding the precision of decimal numbers.

Moreover, the expanded form helps in visualizing the structure of our number system, which is based on powers of 10. Each place value is a power of 10 (1, 10, 100, 1000, etc.), and the expanded form explicitly demonstrates this relationship. This understanding is fundamental for advanced mathematical concepts such as scientific notation and logarithmic scales. In essence, the expanded form is a versatile tool for understanding and manipulating numbers, making it a cornerstone of mathematical literacy.

Applying Expanded Form to the Given Numbers

Now, let's apply the concept of expanded form to the numbers provided in the table. This exercise will reinforce the understanding of place values and how they contribute to the overall value of each number. We will meticulously break down each number into its constituent place values, providing a clear and concise representation of its expanded form. This step-by-step approach will not only help in understanding the expanded form but also in appreciating the structure of the decimal number system.

7,653

To write 7,653 in expanded form, we need to consider each digit's place value:

• The digit 7 is in the thousands place, so its value is 7 x 1000 = 7000.
• The digit 6 is in the hundreds place, so its value is 6 x 100 = 600.
• The digit 5 is in the tens place, so its value is 5 x 10 = 50.
• The digit 3 is in the ones place, so its value is 3 x 1 = 3.

Therefore, the expanded form of 7,653 is 7,000 + 600 + 50 + 3. This representation clearly shows how each digit contributes to the overall value of the number. The 7 in the thousands place contributes 7,000, the 6 in the hundreds place contributes 600, the 5 in the tens place contributes 50, and the 3 in the ones place contributes 3. This breakdown is essential for understanding the magnitude of the number and its constituent parts.

6,745

For the number 6,745, we follow the same process:

• The digit 6 is in the thousands place, so its value is 6 x 1000 = 6000.
• The digit 7 is in the hundreds place, so its value is 7 x 100 = 700.
• The digit 4 is in the tens place, so its value is 4 x 10 = 40.
• The digit 5 is in the ones place, so its value is 5 x 1 = 5.

Thus, the expanded form of 6,745 is 6,000 + 700 + 40 + 5. This expanded form clearly illustrates the value of each digit in the number. The 6 in the thousands place represents six thousand, the 7 in the hundreds place represents seven hundred, the 4 in the tens place represents forty, and the 5 in the ones place represents five. Understanding this breakdown is crucial for performing arithmetic operations and comprehending the structure of the number.

9,751

Now, let's consider the number 9,751:

• The digit 9 is in the thousands place, so its value is 9 x 1000 = 9000.
• The digit 7 is in the hundreds place, so its value is 7 x 100 = 700.
• The digit 5 is in the tens place, so its value is 5 x 10 = 50.
• The digit 1 is in the ones place, so its value is 1 x 1 = 1.

Therefore, the expanded form of 9,751 is 9,000 + 700 + 50 + 1. This representation underscores the significance of each digit's position in determining its value. The 9 in the thousands place is the most significant, contributing nine thousand to the total value, while the 1 in the ones place contributes the least. This understanding is vital for comparing and ordering numbers, as well as for performing complex calculations.

8,990

Next, we will break down the number 8,990:

• The digit 8 is in the thousands place, so its value is 8 x 1000 = 8000.
• The digit 9 is in the hundreds place, so its value is 9 x 100 = 900.
• The digit 9 is in the tens place, so its value is 9 x 10 = 90.
• The digit 0 is in the ones place, so its value is 0 x 1 = 0.

Hence, the expanded form of 8,990 is 8,000 + 900 + 90 + 0, which can also be written as 8,000 + 900 + 90. The inclusion of 0 in the ones place highlights the importance of place value as a placeholder. Even though 0 contributes nothing to the total value, its presence ensures that the other digits occupy their correct positions, maintaining the integrity of the number's magnitude.

3,009

Now, let's express 3,009 in expanded form:

• The digit 3 is in the thousands place, so its value is 3 x 1000 = 3000.
• The digit 0 is in the hundreds place, so its value is 0 x 100 = 0.
• The digit 0 is in the tens place, so its value is 0 x 10 = 0.
• The digit 9 is in the ones place, so its value is 9 x 1 = 9.

Therefore, the expanded form of 3,009 is 3,000 + 0 + 0 + 9, which simplifies to 3,000 + 9. This example further illustrates the role of 0 as a placeholder. The zeros in the hundreds and tens places ensure that the 3 remains in the thousands place and the 9 remains in the ones place. This understanding is particularly important when dealing with numbers that have zeros in non-leading positions.

3,010

Next, we'll write 3,010 in its expanded form:

• The digit 3 is in the thousands place, making its value 3 x 1000 = 3000.
• The digit 0 occupies the hundreds place, so its value is 0 x 100 = 0.
• The digit 1 is in the tens place, with a value of 1 x 10 = 10.
• The digit 0 is in the ones place, contributing 0 x 1 = 0.

Consequently, the expanded form of 3,010 is 3,000 + 0 + 10 + 0, which can be simplified to 3,000 + 10. This number highlights how a digit in the tens place significantly impacts the number's value even when the ones and hundreds places are zero. Understanding this concept is crucial for various mathematical operations and for grasping the relative magnitude of different digits within a number.

8,700

Moving on, let's express 8,700 in expanded form:

• The digit 8 resides in the thousands place, making its value 8 x 1000 = 8000.
• The digit 7 is in the hundreds place, contributing 7 x 100 = 700.
• The digit 0 is in the tens place, so its value is 0 x 10 = 0.
• The digit 0 is in the ones place, with a value of 0 x 1 = 0.

Hence, the expanded form of 8,700 is 8,000 + 700 + 0 + 0, which simplifies to 8,000 + 700. In this instance, the absence of digits in the tens and ones places simplifies the expanded form, emphasizing the value contributed by the thousands and hundreds digits. This kind of simplification is common when dealing with multiples of 100 or 1000, and recognizing these patterns can aid in quicker mental calculations.

2,045

Now, let's write the expanded form for the number 2,045:

• The digit 2 is in the thousands place, so its value is 2 x 1000 = 2000.
• The digit 0 is in the hundreds place, so its value is 0 x 100 = 0.
• The digit 4 is in the tens place, so its value is 4 x 10 = 40.
• The digit 5 is in the ones place, so its value is 5 x 1 = 5.

Thus, the expanded form of 2,045 is 2,000 + 0 + 40 + 5, which simplifies to 2,000 + 40 + 5. This number provides another clear example of the role of zero as a placeholder. The zero in the hundreds place ensures that the digits 2, 4, and 5 maintain their correct place values, contributing to the overall magnitude of the number.

6,107

Finally, let's express the number 6,107 in expanded form:

• The digit 6 occupies the thousands place, thus its value is 6 x 1000 = 6000.
• The digit 1 is in the hundreds place, with a value of 1 x 100 = 100.
• The digit 0 is in the tens place, making its value 0 x 10 = 0.
• The digit 7 is in the ones place, so its value is 7 x 1 = 7.

Consequently, the expanded form of 6,107 is 6,000 + 100 + 0 + 7, which can be simplified to 6,000 + 100 + 7. This final example reinforces the importance of place value and how each digit contributes to the overall value of the number. The presence of zeros in certain place values does not diminish the significance of the other digits; rather, it highlights the structured nature of our number system.

Table of Expanded Forms

To summarize, here is the table with the expanded forms of the numbers:

Conclusion

In conclusion, understanding the expanded form of numbers is a crucial aspect of mathematical literacy. It provides a clear and concise way to represent numbers, highlighting the value of each digit based on its place value. By breaking down numbers into their constituent parts, the expanded form facilitates a deeper understanding of the number system and enhances proficiency in mathematical operations. The examples provided in this article serve as a comprehensive guide, enabling you to confidently express any number in its expanded form. Mastering this concept lays a strong foundation for more advanced mathematical topics and problem-solving skills.