Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Write the number in decimal notation.Charge of an Electron: \(1.602 \times 10^{-19}\) coulomb

Step-by-Step Solution

Answer

After applying the conversion, the charge of electron in decimal notation will be \(0.0000000000000000001602\) coulomb.

Step 1: Understanding the Notation

Firstly, you should understand what scientific notation means. Scientific notation is a way of expressing very large or very small numbers in a compact form, and it is often used in Physics and Mathematics. It has the form \(a \times 10^n\), where \(a\) is a number between 1 and 10 (including 1, but not 10), and \(n\) is an integer.

Step 2: Converting to Decimal Notation

Now, to convert this to decimal notation, one should shift the decimal point of the number \(a\) to the right \(n\) times if \(n\) is positive, and to the left if \(n\) is negative.

Step 3: Applying Conversion

Apply this to our problem: For the charge of an electron, \(1.602 \times 10^{-19}\) coulomb, because \(-19\) is negative, we shift the decimal point in \(1.602\) to the left 19 times. Adding zeroes as placeholders for each shift until there are 19 places after the decimal point.

Key Concepts

Decimal Notation

When we talk about decimal notation, we refer to the representation of numbers using the base of ten. It’s what we commonly use in everyday life. It's the familiar way of writing numbers that consists of digits 0 through 9 and a decimal point.

Converting from scientific notation to decimal notation is a fundamental skill. Scientific notation, such as the example given in the problem, simplifies writing very large or small numbers. To convert a number like \(1.602 \times 10^{-19}\), we move the decimal point of 1.602 nineteen places to the left. This transformation results in a much smaller decimal representation:

• Move the decimal one place for each power of ten.
• For negative powers, move left; for positive powers, move right.
• Use zeroes as placeholders.

After completing this, you end up with: \ 0.0000000000000000001602.

Charge of an Electron

The charge of an electron is a fundamental physics constant often expressed in scientific notation due to its very small size. The elementary charge is approximately \(1.602 \times 10^{-19}\) coulombs. Such a minuscule value is pivotal in atomic and particle physics.

Understanding the charge of an electron helps in gaining insight into:

• How atoms interact and form molecules.
• Electrical conduction in materials.
• The principles behind many modern technologies.

Having an accurate measurement of an electron's charge is essential for calculations involving electromagnetism and quantum mechanics. It depicts how atoms and molecules build up charges that lead to macroscopic fields or forces.

Physics and Mathematics Notation

Using scientific notation is prevalent in both physics and mathematics for several reasons. It provides a clean and efficient way to work with numbers too large or small to be conveniently written in decimal form.

The notation \(a \times 10^n\) is just one way physics and math bridge the gap between complex ideas and simple representations:

• Scientific notation keeps track of significant figures, aiding in precision.
• It simplifies multiplication and division of very large or small numbers.
• Offers a clear view of the scale or magnitude of a number, making comparisons more straightforward.

When using this notation, you improve the accuracy and readability of data, which is crucial in both academic and practical applications, allowing complex problems to be managed more efficiently.