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Chapter 9: Problem 2

Why might a number be written in scientific notation instead of decimal notation?

Short Answer

Expert verified
A number might be written in scientific notation instead of decimal notation to streamline calculation and comparison, especially with extremely large or small numbers. It is more efficient and less prone to errors in such cases.

Step by step solution

01

Understanding Decimal Notation

Decimal notation is the standard system of writing numbers that we use in everyday life. It's also known as base-10 notation because it uses 10 unique digits (0-9). However, when dealing with very large or very small numbers, this notation can become unwieldy.
02

Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is compact and makes computation with these numbers easier. It's written as a product of two parts: a number between 1 and 10; and a power of 10.
03

Comparing the Two Notations

While decimal notation is simpler and more common, scientific notation has specific advantages. It simplifies calculation and comparison, especially with extremely large or small numbers. For example, the distance between two stars in the Universe may be much easier to represent and calculate using scientific notation than decimal notation.

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Key Concepts

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

Decimal Notation
Decimal notation is the way we usually write numbers in our everyday lives. This system, also called base-10, uses ten digits: 0 through 9. Each position in a number holds a place value, which is a power of 10. For example, in the number 345, the digit 5 is in the 'ones' place, the 4 is in the 'tens' place, and the 3 is in the 'hundreds' place.

Decimal notation is very intuitive for writing ordinary numbers. However, when numbers become extremely large or very small, this system can be cumbersome. For instance, writing out 3,000,000,000 can be time-consuming, and keeping track of multiple zeros increases the chance of mistakes.
  • Commonly used for everyday numbers
  • Uses digits 0-9
  • Can be cumbersome for very large or small values
Big Numbers
Big numbers can be difficult to handle in decimal notation because the number of digits can quickly become overwhelming. Scientific notation simplifies the handling of such large numbers by allowing them to be expressed in a more compact form. For instance, the number 300,000,000 can be written in scientific notation as \(3 \times 10^8\).

This form is especially beneficial in scientific calculations where precision and clarity are crucial. By reducing the clutter of zeros, scientific notation emphasizes the most significant digits, and the power of 10 indicates the magnitude of the number.
  • Keeps information concise and clear
  • Makes computation more manageable
  • Shows the order of magnitude efficiently
Small Numbers
Small numbers also present challenges when using decimal notation, just like big numbers. Writing 0.0000005 in decimal form can be inefficient and confusing. By using scientific notation, this number becomes easier to interpret, as it translates to \(5 \times 10^{-7}\).

Scientific notation helps in maintaining accuracy when dealing with very small numbers by focusing on significant digits and the negative power of 10 to signify how small the number is. This makes it much simpler to perform calculations without losing precision.
  • Helps in maintaining accuracy with tiny values
  • Makes small values less prone to error
  • Uses negative powers to define how small a number is
Number Representation
Number representation is all about expressing numbers in different forms to suit a particular context or operation. While decimal notation is great for everyday counts and calculations, scientific notation is invaluable for scientific and engineering tasks.

Scientific notation is especially helpful when a problem involves extreme values. By representing numbers as a combination of a coefficient (a number between 1 and 10) and a power of 10, it minimizes complexity. For example, \(6.022 \times 10^{23}\), Avogadro鈥檚 number, is much easier to work with than 602,200,000,000,000,000,000,000 when performing chemistry calculations.
  • Makes handling extreme numbers feasible
  • Combines significant digits with exponential values
  • Enhances precision in scientific work

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Most popular questions from this chapter

If \(x\) is a nonzero real number, is \(x^{-2}\) always positive, always negative, or positive or negative depending on whether \(x\) is positive or negative? Explain your answer.

Evaluate \(3^{x}\) and \(3^{-x}\) when \(x=-2,-1,0,1\) and 2.

The frequency (in oscillations per second) of a microwave generated by a microwave oven is approximately \(2,450,000,000\) hertz. (One hertz is one oscillation in 1 s.) Write this number in scientific notation.

Multiply. $$(5 a+6)(6 a+5)$$

For Exercises, write in decimal notation. $$7.2 \times 10^{-3}$$
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