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Chapter 1: Problem 31

Review Working It Out 1.1 . Convert the following numbers to scientific notation: a. 7,000,000,000 b. 0.00346 c. 1,238

Short Answer

Expert verified
a. 7 \times 10^9 b. 3.46 \times 10^{-3} c. 1.238 \times 10^3

Step by step solution

01

Identify the first significant digit

For each number, locate the first significant digit.
02

Place decimal after the first significant digit

Rewrite each number with the decimal placed directly after this digit.
03

Determine the exponent for base 10

Count how many places the decimal point has moved from its original position. This number will become the exponent for base 10.
04

Construct the scientific notation

Combine the new decimal number with the base 10 raised to the power of the exponent.

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Key Concepts

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

significant digits
When working with numbers in scientific notation, identifying significant digits is essential. Significant digits, also known as significant figures, are the numbers that carry meaning and contribute to the precision of the figure. For example, in the number 7,000,000,000, the significant digit is '7' because it is the first non-zero number. In 0.00346, the significant digits are '3', '4', and '6'. Notice that leading zeroes are not counted as significant. By recognizing significant digits, we can more accurately express numbers in scientific notation.
  • Always start by identifying the first non-zero digit.
  • Include all following digits up to the desired precision.
base 10 exponent
The base 10 exponent in scientific notation tells us how many places the decimal point has moved. This is crucial for converting large or small numbers into a more manageable form. For instance, to convert 7,000,000,000, we move the decimal point 9 places to the left, making the base 10 exponent 9. Therefore, it becomes 7 × 10^9.
  • Positive exponents indicate large numbers, where the decimal moves left.
  • Negative exponents indicate small numbers, where the decimal moves right.
Understanding the base 10 exponent helps us transform and compare numbers more easily.
decimal placement
Decimal placement is a key step in converting a number to scientific notation. Begin by placing the decimal immediately after the first significant digit. For 7,000,000,000, place the decimal after '7' to form 7.0. In 0.00346, placing the decimal after '3' gives us 3.46. By correctly placing the decimal, you ensure that the number remains accurate and easier to work with.
  • Move the decimal point to immediately follow the first significant digit.
  • Count how many places you moved the decimal to determine your exponent.
number conversion
Converting numbers involves a four-step process. First, locate the first significant digit. Next, place the decimal after this digit. Then, determine the base 10 exponent by counting the places the decimal has moved. Finally, write the number in scientific notation by combining the decimal form with the exponent.
For example, converting 1,238 involves:
  • Identifying '1' as the first significant digit.
  • Placing the decimal after '1' gives us 1.238.
  • The decimal point moved 3 places to the left, so the exponent is 3.
  • Thus, 1,238 becomes 1.238 × 10^3.
Following these steps makes number conversion simple and accurate.

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