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Chapter 5: Problem 101

In Exercises \(91-106\), write each number in scientific notation. $$0.0000202$$

Short Answer

Expert verified
The number \(0.0000202\) in scientific notation is \(2.02 脳 10^{-5}\).

Step by step solution

01

Determine the decimal movement

Firstly, observe the given number \(0.0000202\). To convert this to scientific notation, the decimal point needs to be moved to the right by 5 places so that the number is between 1 and 10.
02

Write the number in scientific notation

The new value of \(a\) that we have after shifting the decimal is 2.02. As the decimal was moved to the right by 5 places, the exponent is -5. So it can be written in scientific notation as \(2.02 脳 10^{-5}\).

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Key Concepts

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

Decimal Movement
When dealing with small or large numbers, we often use scientific notation to simplify them. In this process, one key step involves "decimal movement." Moving the decimal point is crucial because it helps transform the original number into a more convenient form.
To transform a number like 0.0000202 into scientific notation, we can move the decimal point to the right so that the result is a number between 1 and 10. For 0.0000202, you need to move the decimal point five places to the right, which gives you 2.02.
  • This movement of 5 places to the right indicates how the number changes in magnitude.
  • Moving the decimal to the right increases the number by powers of 10.
  • The number of moves determines the value of the exponent, which we will discuss next.
This step simplifies the original number and sets it up for representation in scientific notation.
Exponent
The term "exponent" in scientific notation indicates how many times the decimal point has been moved. It's a small number written above and to the right of a base number, which is always 10 in scientific notation.
For our number 2.02 脳 10^{-5}, the exponent is -5.
  • This tells us the decimal point moved to the right from the original number 0.0000202.
  • Negative exponents represent smaller numbers and show how many positions the decimal has shifted to the right to provide the current form.
This placement and sign of the exponent are essential for correctly interpreting numbers written in scientific notation.
Convert to Scientific Notation
Converting a number to scientific notation is a method of simplifying complex numbers. It's particularly handy for dealing with very large or small numbers efficiently. This process involves two primary steps:
  • First, move the decimal point to get a number between 1 and 10. For 0.0000202, this number is 2.02.
  • Second, note how many places you moved the decimal and in which direction (left for positive exponents and right for negative exponents). In this case, we moved it 5 places to the right, resulting in a negative exponent of -5.
So, 0.0000202 becomes 2.02 脳 10^{-5} in scientific notation.
Scientific notation is thus a tool that allows for compact and consistent representation of numbers that are otherwise awkward to handle.
Small Numbers in Scientific Notation
Representation of small numbers using scientific notation helps in dealing with numbers that are challenging to write and read due to multiple zeros.
For example, small numbers like 0.0000202 are written as 2.02 脳 10^{-5} in scientific notation.
  • This method assists in better understanding and computation.
  • The negative exponent denotes that the decimal moved to the right, indicating a smaller number.
  • It also eliminates errors that can arise due to the miscounting of zeros.
This conversion simplifies communication in science and engineering, where precise, consistent representation of data is crucial.

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