Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Write the number in decimal notation.Width of a Human Hair: \(9.0 \times 10^{-5}\) meter

Step-by-Step Solution

Answer

The width of a human hair described in decimal notation is \(0.00009\) meters.

Step 1: Understanding Scientific Notation

Scientific notation is used to express very large or very small numbers. The number \(9.0 \times 10^{-5}\) is in scientific notation. The base number is 9.0 and the exponent is -5.

Step 2: Convert to Decimal Notation

To convert the scientific notation to decimal notation, start by understanding that \(10^{-5}\) means one should move the decimal in \(9.0\) five places to the left (because the exponent is negative). Each move to the left implies adding a zero, beginning with a decimal point.

Step 3: Result in Decimal Notation

After moving the decimal point five places to the left, the number in decimal notation becomes \(0.00009\). This means that a human hair's width is \(0.00009\) meters in decimal notation.

Key Concepts

Decimal Notation

When we write numbers in decimal notation, we are using the most familiar form of showing numbers that we encounter in daily life. It's built on the base-10 system, which means it revolves around the ten digits from 0 to 9 to represent any number. In decimal notation, the position of each digit is important because it dictates the value of that digit in the context of the number. For instance, in the number 123.45, the '1' is in the hundreds place, meaning it represents '100', while the '5' is in the hundredths place, representing '0.05'.

Understanding decimal notation is crucial for everyday mathematics, from calculating money to measuring distances. It allows us to perform arithmetic operations with ease and is a fundamental concept in various fields, including science and engineering.

Exponents

Exponents are a way to express repeated multiplication of the same number. In the equation a to the power of n, expressed as aⁿ, a is known as the base and n is the exponent or power, and it tells us how many times to multiply the base by itself. For example, 2³ is 2 multiplied by itself 3 times: 2 x 2 x 2 = 8.

Negative Exponents

When exponents are negative, such as 10^-5, this signifies division by the base raised to the opposing positive power. Therefore, 10^-5 means 1 divided by 10⁵ or 0.00001. Negative exponents indicate very small values in scientific notation, helping to simplify the expression of numbers that have many trailing zeros.

Converting Scientific Notation

Scientific notation is a shorthand used to write very large or very small numbers, like the width of a human hair or the distance between stars. It consists of a base number (usually between 1 and 10) and an exponent, which tells you how many places to move the decimal point.

For converting from scientific to decimal notation, follow these steps:

• Identify the exponent. If it's positive, move the decimal point to the right. If negative, move it to the left.
• Move the decimal point as many places as the exponent indicates.
• Add zeros as place holders if necessary, when moving the decimal point past the original digits.

Through this process, we translate the compact form of scientific notation back to a familiar decimal form. Remember, the beauty of scientific notation lies in its ability to simplify and standardize the representation of extremely large or small numbers for ease of reading and calculation.