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Chapter 2: Problem 3

Why is scientific notation useful?

Short Answer

Expert verified
Scientific notation is useful for expressing very large or small numbers in a manageable form, simplifying calculations, and improving clarity in communication across scientific disciplines.

Step by step solution

01

Understanding the Purpose of Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is used to make such numbers more manageable and easier to compare.
02

Examining the Structure

Scientific notation is structured as the product of a number between 1 and 10 and a power of 10. This makes it easier to perform calculations like multiplication and division, as you can simply add or subtract the exponents.
03

Recognizing Its Role in Various Fields

Scientific notation is used across various fields such as physics, chemistry, and engineering, to accurately and concisely represent measurements, perform calculations, and communicate findings.

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Key Concepts

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

Understanding the Purpose of Scientific Notation
Have you ever wondered why scientists prefer using scientific notation, particularly when dealing with extremely large or tiny numbers? Scientific notation plays a crucial role in ensuring that these types of numbers are manageable. It provides a standardized method for expressing such numbers, enabling easier manipulation and clear communication. For instance, the distance to the nearest star is about 40,208,000,000,000 kilometers, which becomes a more manageable 4.0208 x 1013 kilometers in scientific notation. This use of scientific notation simplifies comparisons and calculations with large numbers, saving time and reducing the risk of errors.
Structuring Numbers Using Scientific Notation
Scientific notation is carefully structured to improve clarity and consistency. It is represented as the product of two parts: a coefficient and a power of 10. The coefficient is a number between 1 and 10, and it is multiplied by 10 raised to an exponent. The exponent indicates the number of places the decimal point has been moved. For example, the mass of an electron, approximately 0.00000000000000000000000000000091093822 kg, would be written in scientific notation as 9.1093822 x 10-31 kg. When working with scientific notation, operations like multiplication and division are much simpler, as they usually involve combining exponents, making the math less cumbersome and more intuitive.
Scientific Notation Across Various Fields
The use of scientific notation extends beyond just mathematics鈥攊t is a universal language across multiple scientific disciplines. In physics, it helps in expressing the vast ranges of forces, distances, and timescales. For example, the Planck constant is neatly expressed as 6.626 x 10-34 m2kg/s. Chemists use it to measure substance amounts in moles, while engineers might use it for calculations related to electrical currents or civil structures. Moreover, astronomers employ scientific notation to articulate the enormous distances between celestial bodies with ease. Therefore, scientific notation is not just about compact representation but also about seamless communication, enabling scientists and engineers worldwide to exchange information without confusion. In essence, it is the epitome of precision and efficiency in scientific discourse.

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Most popular questions from this chapter

In 1999, scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun but occupying less space than our moon. Suppose that one of these black holes has a mass of \(1 \times 10^{3}\) suns and a radius equal to one-half the radius of our moon. What is its density in grams per cubic centimeter? The mass of the sun is \(2.0 \times 10^{30} \mathrm{~kg}\), and the radius of the moon is \(2.16 \times 10^{3} \mathrm{mi}\). (Volume of a sphere \(=\frac{4}{3} \pi r^{3}\).)

Correct any answers that have the incorrect number of significant figures. (a) \((908.4-3.4) \div 3.52 \times 10^{4}=0.026\) (b) \((1206.7-0.904) \times 89=1.07 \times 10^{5}\) (c) \((876.90+98.1) \div 56.998=17.11\) (d) \((4.55 \div 407859)+1.00098=1.00210\)

Round the number on the left to the number of significant figures indicated by the example in the first row. (Use scientific notation as needed to avoid ambiguity.) \begin{tabular}{lccc} & Rounded to 4 Significant Figures & Rounded to 2 Significant Figures & Rounded to 1 Significant \\ \(94.52118\) & \(94.52\) & 95 & Figure \\ \(105.4545\) & & & \\ \(0.455981\) & & & \\ \(0.009999991\) & & & \\ \hline \end{tabular}

Perform each calculation to the correct number of significant figures. (a) \(89.3 \times 77.0 \times 0.08\) (b) \(\left(5.01 \times 10^{5}\right) \div\left(7.8 \times 10^{2}\right)\) (c) \(4.005 \times 74 \times 0.007\) (d) \(453 \div 2.031\)

A supposedly gold crown is tested to determine its density. It displaces \(10.7 \mathrm{~mL}\) of water and has a mass of \(206 \mathrm{~g}\). Could the crown be made of gold?
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