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Chapter 3: Problem 5

Write the following numbers in scientific notation. $$ 179,000,000,000,000,000,000 $$

Short Answer

Expert verified
Question: Convert the number 179,000,000,000,000,000,000 into scientific notation. Answer: The number 179,000,000,000,000,000,000 can be written in scientific notation as 1.79 脳 10^20.

Step by step solution

01

Steps to Convert the Number into Scientific Notation

Follow the steps below to write the given number in scientific notation:
02

Find the Decimal Point's New Position

We need to move the decimal point after the first significant digit, which is 1 in this case. $$ 179,000,000,000,000,000,000 -> 1.79 $$
03

Count Decimal Point Movement

Count how many places the decimal point moved from its initial position. In this case, it moved 20 places to the left. $$ 179,000,000,000,000,000,000 $$ ^ (20 places)
04

Write in Scientific Notation

Now, write the number in the form "a x 10^n", where "a" is the number determined in step 1 (1.79), and "n" is the number of places the decimal point moved (20). $$ 1.79 \times 10^{20} $$ The given number in scientific notation is: $$ 1.79 \times 10^{20} $$

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Key Concepts

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

Understanding the Decimal Point
The decimal point is a crucial part of numbers as it separates the whole number from the fractional part. When writing a number in scientific notation, you need to shift the decimal point to form a new number between 1 and 10. This involves moving the point until it's right after the first non-zero digit, making it easier to manage large or very small numbers.
For example, in the original number 179,000,000,000,000,000,000, the decimal is at the end since it represents a whole number. To convert this into scientific notation, we move the decimal 19 spaces to the left, creating 1.79.
The Role of Significant Figures
Significant figures are the digits in a number that contribute to its precision. When converting numbers into scientific notation, maintaining these significant figures is essential, as they indicate the accuracy of your number. In practice, significant figures include all non-zero digits, any zeros between them, and any trailing zeros in the decimal portion.
For our number 179,000,000,000,000,000,000, only the digits 179 are significant. Thus, when it's expressed in scientific notation, it becomes 1.79, which keeps the precision of the original number intact.
The Importance of the Exponent
In scientific notation, the exponent tells you how many times you must multiply the base number by 10 to get back to the original number. Once you've repositioned the decimal point, how far you've moved it will be represented as the exponent of 10.
For example, moving the decimal in 179,000,000,000,000,000,000 from the end to just after the first digit gives us an exponent of 19. This means you multiply 1.79 by 10 raised to the 19th power to retrieve the original number. This representation makes it easier to read and work with extremely large or small values.
Simplifying Large Numbers
Scientific notation is particularly useful for handling large numbers, which can be cumbersome and error-prone when written out completely. Such numbers often appear in scientific and engineering contexts, where precision and clarity are paramount.
By using scientific notation, very large numbers like 179,000,000,000,000,000,000 can be condensed to a concise format like 1.79 x 10^19. This format shows all necessary information at a glance, making calculations more efficient and reducing the likelihood of errors.

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