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Chapter 1: Problem 97

Briefly describe each of the following ideas: (a) SI base units; (b) significant figures; (c) natural law; (d) exponential notation.

Short Answer

Expert verified
The SI base units are a set of seven fundamental units used globally for measurement, significant figures denote the precision of a measurement, a natural law is a mathematical model describing a pattern in nature, and exponential notation is a way of expressing very large or very small numbers.

Step by step solution

01

SI Base Units

SI base units are the seven fundamental units in the International System of Units, from which all other units can be derived. They include the meter for length, kilogram for mass, second for time, ampere for electric current, kelvin for thermodynamic temperature, mole for amount of substance, and candela for luminous intensity.
02

Significant Figures

Significant figures are the digits in a number that carry meaningful information about its precision. They include all certain numbers plus one uncertain number, which is often the last digit in the measurement.
03

Natural Law

A natural law is a description, in the form of a mathematical model, of a pattern that nature follows under certain conditions. Examples include Newton's law of motion or the law of gravity.
04

Exponential Notation

Exponential notation is a way of writing numbers that accommodates values too large or too small to be conveniently written in standard decimal form. It represents numbers as a product of two parts: a number between 1 and 10, and a power of 10. For instance, \(2.3 \times 10^4\) is an example of exponential notation.

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Key Concepts

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

SI Base Units
The SI base units are fundamental to our understanding of measurements in science. Developed by the International System of Units, these seven units form the foundation for all other measurements. Each unit corresponds to a specific physical quantity essential for scientific experiments and daily applications:
  • The meter (m) measures length.
  • The kilogram (kg) assesses mass.
  • The second (s) defines time.
  • The ampere (A) quantifies electric current.
  • The kelvin (K) evaluates thermodynamic temperature.
  • The mole (mol) counts the amount of substance.
  • The candela (cd) gauges luminous intensity.
These units allow scientists and engineers around the world to communicate in a common language, ensuring accuracy and repeatability in their results. Understanding these base units is crucial for anyone entering a field that involves scientific measurements.
Significant Figures
Significant figures are vital in representing the precision of a measurement. They convey the reliability of a measured quantity, indicating which digits are known with certainty and which are estimated. Here's how to recognize significant figures:
  • All non-zero digits are always significant. For example, in 345, all three digits are significant.
  • Any zeros between significant digits are also regarded as significant. In 3007, all four digits are significant.
  • Leading zeros do not count as significant. So, 0.0045 has two significant figures.
  • Trailing zeros in a number containing a decimal point are significant. For example, 45.00 has four significant figures.
Knowing how to determine significant figures helps in maintaining the integrity of calculations, ensuring that your results are as precise as your measurements permit.
Natural Law
Natural laws are foundational principles that describe predictable behaviors in the universe. They are represented using mathematical models, which help scientists understand natural patterns under defined conditions. Some key aspects include:
  • Natural laws remain constant and universal over time. This timelessness allows them to be a reliable basis for scientific study.
  • Examples of natural laws include Newton's laws of motion or the law of gravitation, which describe how objects move with forces acting upon them.
  • Natural laws form the basis of theoretical predictions, helping scientists explain why certain phenomena occur in a repeatable manner.
Exploring these laws allows us to make sense of our world, predicting future events based on the unchanging nature of the laws discovered through observation and experimentation.
Exponential Notation
Exponential notation simplifies the expression of very large or very small numbers, making them easier to read and work with. The notation consists of two parts: a coefficient and a power of 10. For example, the number 2,300 can be written as \(2.3 \times 10^3\).
Key elements of understanding exponential notation include:
  • The coefficient is a number between 1 and 10; it provides the significant figures of the number.
  • The exponent tells you how many times to multiply the coefficient by 10. A positive exponent indicates a large number, while a negative exponent signifies a small fraction.
  • Using exponential notation helps in performing calculations more efficiently, especially with calculators or computer software that can handle powers of ten directly.
Grasping the concept of exponential notation is essential in fields like science and engineering, where dealing with extremely large or small numbers is commonplace.

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Most popular questions from this chapter

To determine the volume of an irregularly shaped glass vessel, the vessel is weighed empty \((121.3 \mathrm{g})\) and when filled with carbon tetrachloride (283.2 g). What is the volume capacity of the vessel, in milliliters, given that the density of carbon tetrachloride is \(1.59 \mathrm{g} / \mathrm{mL} ?\)

Express the result of each of the following calculations in exponential form and with the appropriate number of significant figures. (a) \(\left(4.65 \times 10^{4}\right) \times\left(2.95 \times 10^{-2}\right) \times\left(6.663 \times 10^{-3}\right) \times 8.2=\) (b) \(\frac{1912 \times\left(0.0077 \times 10^{4}\right) \times\left(3.12 \times 10^{-3}\right)}{\left(4.18 \times 10^{-4}\right)^{3}}=\) {c} \(\left(3.46 \times 10^{3}\right) \times 0.087 \times 15.26 \times 1.0023=\) (d) \(\frac{\left(4.505 \times 10^{-2}\right)^{2} \times 1.080 \times 1545.9}{0.03203 \times 10^{3}}=\) (e) \(\frac{\left(-3.61 \times 10^{-4}\right)+\sqrt{\left(3.61 \times 10^{-4}\right)^{2}+4(1.00)\left(1.9 \times 10^{-5}\right)}}{2 \times(1.00)}\) [Hint: The significant figure rule for the extraction of a root is the same as for multiplication.]

Express each value in exponential form. Where appropriate, include units in your answer. (a) speed of sound (sea level): 34,000 centimeters per second (b) equatorial radius of Earth: 6378 kilometers (c) the distance between the two hydrogen atoms in the hydrogen molecule: 74 trillionths of a meter (d) \(\frac{\left(2.2 \times 10^{3}\right)+\left(4.7 \times 10^{2}\right)}{5.8 \times 10^{-3}}=\)

For a solution containing \(6.38 \%\) para-diclorobenzene by mass in benzene, the density of the solution as a function of temperature ( \(t\) ) in the temperature range 15 to \(65^{\circ} \mathrm{C}\) is given by the equation \(d(\mathrm{g} / \mathrm{mL})=1.5794-1.836 \times 10^{-3}(t-15)\) At what temperature will the solution have a density of \(1.543 \mathrm{g} / \mathrm{mL} ?\)

In the third century \(\mathrm{BC}\), the Greek mathematician Archimedes is said to have discovered an important principle that is useful in density determinations. The story told is that King Hiero of Syracuse (in Sicily) asked Archimedes to verify that an ornate crown made for him by a goldsmith consisted of pure gold and not a gold-silver alloy. Archimedes had to do this, of course, without damaging the crown in any way. Describe how Archimedes did this, or if you don't know the rest of the story, rediscover Archimedes's principle and explain how it can be used to settle the question.
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