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Chapter 3: Problem 6

Write the following numbers in scientific notation. $$ 100,000 $$

Short Answer

Expert verified
Question: Express the number 100,000 in scientific notation. Answer: 1.0 脳 10^5

Step by step solution

01

Identify the decimal point placement

First identify the current placement of the decimal point in the number, which is at the end of the large number (100,000.). Then, we need to move the decimal point so that we have a number between 1 and 10.
02

Move the decimal point to the correct position

Move the decimal point 5 places to the left so that it sits just after the first non-zero digit (1). The new number would be 1.0, which is between 1 and 10 (including 1 but excluding 10). This is our value for \(a\).
03

Determine the exponent

The exponent (\(b\)) represents the number of places the decimal point was moved. In this case, we moved it 5 places to the left, so \(b = 5\).
04

Write the number in scientific notation

Using the values of \(a\) and \(b\) we found in the previous steps, we can now write the given number in scientific notation: $$ 1.0 \times 10^5 $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Decimal Point Placement
The concept of decimal point placement is crucial when working with scientific notation. In a large number like 100,000, although the decimal point is not explicitly written, it's essential to understand that it is at the very end of the number. This is because every whole number can be written with a decimal, like 100,000.0.

When converting a number to scientific notation, we start by moving the decimal point to create a new number between 1 and 10. This involves shifting the decimal to lie after the first non-zero digit of the number.
For instance, moving the decimal 5 places to the left gives us 1.0. This step ensures that the number is now appropriately formatted for scientific notation.
From Large Numbers to Scientific Notation
Number conversion is the process of changing a typical number format into scientific notation by moving the decimal point to obtain a new number that is easy to work with.

Starting with a large number like 100,000, converting to scientific notation simplifies working with it in mathematical operations.
  • First, acknowledge where the decimal starts; here, it's assumed to be at the end: 100,000 is the same as 100,000.0.
  • Shift the decimal left to form a number between 1 and 10. For 100,000, moving the decimal 5 places left results in 1.0.

Transforming to a consistent format makes complex calculations between vastly different numbers manageable.
The Role of the Exponent in Scientific Notation
The exponent in scientific notation indicates how many places the decimal point moved. It is crucial as it determines the scale or size of the original number after conversion.

In our example, after moving the decimal 5 positions left to convert 100,000 to 1.0, the exponent becomes 5.

This is specified in the scientific notation formula: \[ a \times 10^b \]where \(a\) is the significant figure and \(b\) is the exponent. Thus, 100,000 becomes \[ 1.0 \times 10^5 \].
  • The 5 indicates that the number is multiplied by 10 raised to the power of 5, effectively reflecting its original size.
  • Positive exponents are used for numbers greater than one, while negative exponents apply if the number is less than one.
Understanding the exponent's role helps manage numbers' magnitude across different contexts.

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Most popular questions from this chapter

For the following problems, convert the numbers from scientific notation to standard decimal form. A car traveling at an average speed will travel a distance about equal to the length of the smallest fingernail in \(3.16 \times 10^{-4}\) seconds.

For the following exercises, perform the indicated operations. $$ -0.012-(-0.111)-(0.035) $$

Convert the numbers used in the following problems to scientific notation. A subatomic particle called a neutral pion has a half-life of about 0.0000000000000001 second.

Convert the numbers used in the following problems to scientific notation. \(\begin{array}{llllllll}\text { There is an } & \text { irregularly shaped galaxy, } & \text { named } & \text { NGC } & 4449, & \text { that is } & \text { about }\end{array}\) 250,000,000,000,000,000,000,000 meters from earth.

Write the following numbers in scientific notation. $$ 87,000,000 $$
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