Problem 50 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. Near the surface of the earth, the speed of sound is 1195 feet per second.

Step-by-Step Solution

Answer

Question: Convert the speed of sound near the surface of the Earth, 1195 feet per second, into scientific notation. Answer: 1.195 x 10^3 feet per second

Step 1: Identify the given number

The given number is the speed of sound near the surface of the Earth, which is 1195 feet per second.

Step 2: Write the number in decimal form

Currently, the number is written as 1195. To write it in decimal form, we need to find the position after first non-zero digit and add a decimal point. Hence, we have 1.195

Step 3: Count the number of positions the decimal point was moved

We moved the decimal point from its original position (after the 5) to its new position (after the 1). This represents a total of 3 positions to the left.

Step 4: Rewrite the number in scientific notation

Now we can rewrite the number in scientific notation. We'll use the decimal form of the number (1.195) and multiply it by 10 raised to the power of the number of positions the decimal point was moved (3): 1.195 \times 10^3

Step 5: Final Answer

The speed of sound near the surface of the Earth, given as 1195 feet per second, can be expressed in scientific notation as 1.195 \times 10^3 feet per second.

Key Concepts

Converting to Scientific Notation

When working with very large or very small numbers, scientific notation becomes an invaluable tool for simplifying calculations and comparison. Converting a number to scientific notation involves creating a product of a number between 1 and 10 and a power of 10.

For instance, if we take the number 1195, the process begins with placing the decimal point such that only one non-zero digit is before it, resulting in 1.195. Next, determine the power of 10 by counting how many places the decimal moves from the original number to the placement in the new form. In this case, it moves three places, creating a power of 10, which is then expressed as an exponent. The correctly converted scientific notation of 1195 is therefore \(1.195 \times 10^3\).

Decimal Form Conversion

Conversely, converting from scientific notation to decimal form can sometimes be a challenge if the exponent is large. The key is to focus on the power of 10. If the exponent is positive, move the decimal point to the right; if it’s negative, move it to the left.

For the scientific notation \(1.195 \times 10^3\), move the decimal to the right three places, giving us 1195, the decimal form. Remember, every step to the right is a multiplication by 10. Pay attention to the initial position of the decimal point in the original number to avoid mistakes in the conversion.

Exponents in Scientific Notation

The exponent in scientific notation explicitly shows the magnitude of a number. Proper use of exponents requires understanding positive and negative powers. If the original number is large, the exponent will be positive, as you're counting the number of times you’d multiply by 10. Small numbers, meanwhile, will have a negative exponent, reflecting the number of divisions by 10.

For example, an exponent of 3, as in \(10^3\), signifies three tens multiplied together (1000). The exponent provides a shorthand to describe magnitude without writing out extended zeros, making calculations with varying scales more manageable.

Magnitude Representation

Scientific notation elegantly conveys the magnitude of a number. It allows for quick assessment of scale, especially useful in fields like astrophysics, microbiology, and chemistry, where numbers can range from the incredibly vast to the infinitesimally small.

In essence, the exponent indicates how many times the number is to be multiplied or divided by 10 to reach its true size. A positive exponent signifies a large magnitude, while a negative exponent indicates a much smaller number. Through scientific notation, comparisons between different orders of magnitude become far simpler, enabling easier arithmetic and clearer visualization of relative sizes.