Vocabulary Terms
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What number is our number system based on?
- Our number system is based on 10, which means it is a base-10 system (also known as the decimal system). In this system, each digit in a number has a place value that is a power of 10.
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Expanded Form
- Expanded form is a way of writing a number that shows the value of each digit. In expanded form, numbers are broken down into sums of each digit multiplied by its place value.
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Place Value
- Place value refers to the value of a digit depending on its position in a number. In the number 825, for example, the digit 8 is in the hundreds place, the digit 2 is in the tens place, and the digit 5 is in the ones place.
Problems
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Write the expanded form with base 10 exponent notation:
- 825.29
- Expanded Form:
- \(8 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 2 \times 10^{-1} + 9 \times 10^{-2}\)
- Work shown:
- \(8 \times 100 + 2 \times 10 + 5 \times 1 + 2 \times 0.1 + 9 \times 0.01\)
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Write the expanded form with base 10 exponent notation:
- 5.3257
- Expanded Form:
- \(5 \times 10^0 + 3 \times 10^{-1} + 2 \times 10^{-2} + 5 \times 10^{-3} + 7 \times 10^{-4}\)
- Work shown:
- \(5 \times 1 + 3 \times 0.1 + 2 \times 0.01 + 5 \times 0.001 + 7 \times 0.0001\)
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Write the expanded form with base 10 exponent notation:
- 0.45
- Expanded Form:
- \(4 \times 10^{-1} + 5 \times 10^{-2}\)
- Work shown:
- \(4 \times 0.1 + 5 \times 0.01\)
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Write the expanded form with base 10 exponent notation:
- 125.325
- Expanded Form:
- \(1 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 3 \times 10^{-1} + 2 \times 10^{-2} + 5 \times 10^{-3}\)
- Work shown:
- \(1 \times 100 + 2 \times 10 + 5 \times 1 + 3 \times 0.1 + 2 \times 0.01 + 5 \times 0.001\)
Explanation
Why are numbers in expanded form with base 10 notation equal to the number itself for our number system?
Numbers in expanded form with base 10 notation are equal to the number itself because each component of the expanded form represents the actual value contributed by that digit based on its position.
For example, in the number 825.29, breaking it down into \(8 \times 10^2\) (which is 800), \(2 \times 10^1\) (which is 20), \(5 \times 10^0\) (which is 5), \(2 \times 10^{-1}\) (which is 0.2), and \(9 \times 10^{-2}\) (which is 0.09) allows us to sum these values (800 + 20 + 5 + 0.2 + 0.09 = 825.29). Each digit's position indicates its value multiplied by a power of ten, which directly shows how it contributes to the overall number. In our base-10 system, every number can be expressed as a sum of its parts, revealing why expanded form is equivalent to the original number.
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