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Chapter 2: Problem 15

How would the number \(6.31 \times 10^{17}\) be represented in \(\mathrm{E}\) notarion?

Short Answer

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Question: Represent the number \(6.31 \times 10^{17}\) in E notation. Answer: \(6.31\mathrm{E}17\)

Step by step solution

01

Identify the mantissa and exponent

In the given number, \(6.31 \times 10^{17}\), the mantissa is 6.31 and the exponent is 17.
02

Convert scientific notation to E notation

To write the number in E notation, we need to represent the exponent using an E (capitalized). So, we will replace the "x 10" part with an "E" and keep the same exponent of 10. The E notation of the given number is: \(6.31\mathrm{E}17\).

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Key Concepts

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

E Notation
E Notation is a way of writing numbers that are too large or too small for standard decimal notation. It is frequently used in computer science and scientific fields because it simplifies reading and writing such numbers. The format involves writing a number in the form of a mantissa, followed by an "E", and then an exponent. The letter "E" stands for exponent, and it indicates "times ten raised to the power of".
  • For instance, the number 6.31E17 means 6.31 multiplied by 10 to the power of 17.
  • This method helps quickly understand the scale of the number without writing out numerous zeros.
This notation is particularly useful in programming environments and databases that can't easily handle very large numbers in decimal form, providing a clear and concise way to express large values.
Exponential Notation
Exponential Notation refers to a way of expressing numbers that accommodates the vast range of possible values in mathematics and science. Exponential notation makes use of powers of ten to indicate the size of the number. This notation is very similar to scientific and E notation but typically emphasizes the mathematical relationship more.
  • The typical exponential notation is written as a number (the mantissa) multiplied by ten to some power (the exponent).
  • This allows a consistent way to easily grasp the magnitude of numbers both large and small.
  • For instance, in Exponential Notation, 6.31 x 1017 provides a succinct way to display the value.
Rather than writing out 631 followed by 15 zeros, exponential notation offers a way to express this very succinctly. This helps in minimizing error, offering clarity and reducing space.
Mantissa and Exponent
In scientific, E, and exponential notation, two key components define how numbers are expressed: the mantissa and the exponent. The mantissa is the significant part of the number. It consists of the digits that make up its precision, separated from the exponential factor.
  • For example, in the number 6.31 x 1017, 6.31 is the mantissa.
  • The choice of mantissa depends on the number of significant digits you wish to retain.
The exponent indicates the power of ten by which the mantissa should be multiplied.
  • In 6.31 x 1017, 17 is the exponent, indicating the mantissa should be multiplied by ten 17 times.
  • This reflects the actual size and scale of the number like how many places the decimal point will move.
Together, the mantissa and exponent provide a robust way of simplifying large or small numeric values, making them easily readable and workable in calculations and algorithms.

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