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Chapter 5: Problem 105

In Exercises \(91-106\), write each number in scientific notation. $$3.14159$$

Short Answer

Expert verified
The scientific notation of 3.14159 is \(3.14159 \times 10^0\).

Step by step solution

01

Identify the number to be written in scientific notation

The number given is 3.14159.
02

Write the number in the form \((a \times 10^n)\

In this case, since the given number is between 1 and 10, it is its own coefficient, and since it's not been moved in the process, the exponent value of 10 is 0. Therefore we have: \(3.14159 \times 10^0\)

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Key Concepts

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

Understanding Exponents
In mathematics, an exponent represents the power to which a number, called the base, is raised. An exponent is a small number placed to the upper right of the base number. For example, in the expression \(10^3\), 10 is the base, and 3 is the exponent. This means that you multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\). Thus, exponents are effectively shorthand for repeated multiplication. They help us express large or small numbers in a more readable form, particularly in scientific notation.
The Role of the Coefficient
In scientific notation, the coefficient is the number that is multiplied by a power of 10. It is usually a number between 1 and 10. In the format \(a \times 10^n\), 'a' is the coefficient.
  • For example, in \(3.14159 \times 10^0\), \(3.14159\) is the coefficient.
  • You will notice that the coefficient maintains the significant figures of the original number, which is crucial for precision.
The process of forming a coefficient involves shifting the decimal point in your original number so that it falls between 1 and 10, though in this case, no shifting was necessary.
Decoding Mathematical Notation
Mathematical notation comprises the symbols and signs used to express numbers and operations compactly. Scientific notation is a form of mathematical notation. It provides a way to write very large or very small numbers succinctly using powers of ten. When you see a number like \(3.14159 \times 10^0\), the notation tells us:
  • '\(3.14159\)' is the main number, capturing all significant data.
  • '\(\times 10^0\)' indicates that the number is not scaled up or down, retaining its original size.
The beauty of mathematical notation lies in its ability to convey complex ideas clearly and efficiently, which is why it is vital for the sciences.
Step-by-Step Approach to Algebra
Step-by-step algebra is a methodical approach to solving algebraic expressions and equations. It involves breaking down the problem into manageable steps:
  • Identify key numbers or variables involved in the problem, as seen in Step 1 of the solution where the number \(3.14159\) was identified.
  • Apply relevant algebraic rules or formulas to those numbers, like converting the number into scientific notation by recognizing its coefficient and exponent.
  • Ensure each step logically follows and supports the previous one to maintain consistency within the solution.
  • Verify the final result to ensure accuracy, as demonstrated when confirming \(3.14159 \times 10^0\) represents the number in scientific notation correctly since it required no decimal place shift.
Following a step-by-step approach helps to simplify complex algebraic problems, making them more approachable.

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