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Magazine Chapter 1: Problem 33
Write each of the following numbers in expanded form. $$30,068$$
Short Answer
Expert verified
30,068 in expanded form is 30,000 + 60 + 8.
Step by step solution
01
Identify the Place Values
First, recognize the place value of each digit in the number \(30,068\). The number has 5 digits, so from left to right: 3 is in the 'ten-thousands' place, 0 is in the 'thousands' place, another 0 is in the 'hundreds' place, 6 is in the 'tens' place, and 8 is in the 'ones' place.
02
Express Each Digit with its Place Value
Convert each digit into a number that represents its place value. For each digit in \(30,068\):- The digit 3 is in the ten-thousands place, equaling \(3 \times 10,000 = 30,000\).- The digit 0 is in the thousands place, equaling \(0 \times 1,000 = 0\).- The digit 0 is in the hundreds place, equaling \(0 \times 100 = 0\).- The digit 6 is in the tens place, equaling \(6 \times 10 = 60\).- The digit 8 is in the ones place, equaling \(8 \times 1 = 8\).
03
Write the Number in Expanded Form
Combine the expanded forms of all the digits into one expression. Thus, the number \(30,068\) in expanded form is:\[30,000 + 0 + 0 + 60 + 8\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Place Value
Place value is a fundamental concept in mathematics that helps us understand the value of each digit in a number based on its position. Each position in a number has a different place value, and this becomes particularly important when dealing with larger numbers, such as five-digit numbers like 30,068.
In a five-digit number:
In a five-digit number:
- The digit farthest to the right is in the 'ones' place. This position has a place value of 1.
- The next digit is in the 'tens' place, with a place value of 10.
- Following is the 'hundreds' place, which has a place value of 100.
- The 'thousands' place comes next, with a place value of 1,000.
- Finally, there's the 'ten-thousands' place, which has the highest place value in a five-digit number, 10,000.
Exploring Five-Digit Numbers
Five-digit numbers consist of digits ranging from the position of the 'ones' place to the 'ten-thousands' place. A five-digit number, like 30,068, tells us not only about quantity but also about the arrangement that provides its value.
As the size of numbers increases, place values become more significant:
As the size of numbers increases, place values become more significant:
- A number in the "ten-thousands" position substantially increases the overall value of the number. In 30,068, the digit 3 is in the ten-thousands place.
- The presence of 0 in other positions means these digits contribute nothing to the actual value except as placeholders.
- The "tens" and "ones" places still matter, as they fine-tune the number's exactness.
Understanding Mathematical Expressions
Mathematical expressions are used to represent numbers in different ways, helping us to see the construction of a number using its place values. An expanded form is a type of mathematical expression that breaks down numbers into sums of each digit multiplied by its matching place value.
For example, in the expanded form of 30,068:
This approach illustrates the number's make-up mathematically. Expressing numbers in expanded form offers a clear and visual way to comprehend how different operations create a complete number, adding depth to numerical understanding.
For example, in the expanded form of 30,068:
- The digit 3 is multiplied by its place value of 10,000, resulting in 30,000.
- Each 0 in the thousands and hundreds places shows its need for multiplication by 1,000 or 100, giving a result of 0.
- The digit 6, representing the tens place, is multiplied by 10, yielding 60.
- The digit 8, in the ones place, maintains its identity by multiplying it by 1, to contribute a simple 8.
This approach illustrates the number's make-up mathematically. Expressing numbers in expanded form offers a clear and visual way to comprehend how different operations create a complete number, adding depth to numerical understanding.
