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Chapter 9: Problem 98

SCIENTIFIC NOTATION Rewrite the number in scientific notation. $$3,984,328$$

Short Answer

Expert verified
The number 3,984,328 in scientific notation is \(3.984328 * 10^6\)

Step by step solution

01

Identify the digits and place

The number to be converted into scientific notation is 3,984,328. The first digit is 3, and considering the place value it represents, it's in the millions place.
02

Construct the preliminary scientific notation

Place the decimal point after the first non-zero digit, which gives \(3.984328\). Because we moved six places to the right, we apply '10' raised to a positive 6, denoted as \(10^6\).
03

Final Scientific notation

The number in scientific notation is hence written as: \(3.984328 * 10^6\)

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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 an essential concept in mathematics, especially when dealing with large numbers and scientific notation. Each digit in a number has a specific value, depending on its position within the number. For instance, in the number 3,984,328:
  • The '3' is in the millions place, which means it represents 3 million.
  • The '9' represents 900,000 as it is in the hundred thousand place.
  • Similarly, each subsequent digit represents decreasing powers of ten, down to the '8' in the units place.
Understanding this helps to determine how significant a digit is relative to the others and aids in conversion to scientific notation.
Role of the Decimal Point
The decimal point is an essential component in scientific notation. It serves not just to separate whole numbers from fractions but also to indicate precision when converting a number. In transforming the number 3,984,328 into scientific notation, the decimal point is moved to just after the first non-zero digit, yielding 3.984328.

This positioning ensures the number is normalized and written in a form where one digit stands before the decimal, making it easier to handle mathematically across scales. The placement of the decimal point also guides how we count the power of ten used in scientific notation.
The Power of Exponents
Exponents play a crucial role in scientific notation. They indicate how many times a number should be multiplied by ten. When converting 3,984,328 to scientific notation, we move the decimal point six places to the left to position it after the first number. Hence, the number is expressed as 3.984328 multiplied by 10 raised to the power of 6, or \[10^6\].

Here, the exponent 6 means the number 3.984328 should be multiplied by 1 million (or 10 raised to the 6th power). This way, large values are compactly represented and easily manageable, which is essential in fields like science and engineering.
Importance of Non-Zero Digit
The concept of a non-zero digit is pivotal to expressing numbers in scientific notation. A non-zero digit is any digit from 1 to 9, and it forms the foundation of the notation system. In our example, we choose '3' because it is the first non-zero digit at the start of 3,984,328.

This choice influences where the decimal point is placed in scientific notation, ensuring the expression is as efficient as possible. The presence of non-zero digits prevents redundancy and maintains clarity, crucial for understanding and processing the information represented by large numbers.

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