Question

What exponential notation do the following abbreviations represent? (a) d, (b) c, (c) \(f\), (d) \(\mu\), (e) \(M\), (f) \(k\), (g) \(n\), (h) \(m\), (i) p.

Short answer

(a) The abbreviation "d" represents deci (\(10^{-1}\)). (b) The abbreviation "c" represents centi (\(10^{-2}\)). (c) The abbreviation "f" represents femto (\(10^{-15}\)). (d) The abbreviation "μ" represents micro (\(10^{-6}\)). (e) The abbreviation "M" represents mega (\(10^{6}\)). (f) The abbreviation "k" represents kilo (\(10^{3}\)). (g) The abbreviation "n" represents nano (\(10^{-9}\)). (h) The abbreviation "m" represents milli (\(10^{-3}\)). (i) The abbreviation "p" represents pico (\(10^{-12}\)).

Step-by-step solution

(a) d

The abbreviation "d" represents the prefix "deci," which means a factor of one-tenth or \(\frac{1}{10}\). In exponential notation, this is represented as \(10^{-1}\).

(b) c

The abbreviation "c" represents the prefix "centi," which means a factor of one-hundredth or \(\frac{1}{100}\). In exponential notation, this is represented as \(10^{-2}\).

(c) f

The abbreviation "f" represents the prefix "femto," which means a factor of one quadrillionth or \(\frac{1}{1,000,000,000,000,000}\). In exponential notation, this is represented as \(10^{-15}\).

(d) μ

The abbreviation "μ" represents the prefix "micro," which means a factor of one millionth or \(\frac{1}{1,000,000}\). In exponential notation, this is represented as \(10^{-6}\).

(e) M

The abbreviation "M" represents the prefix "mega," which means a factor of one million or \(1,000,000\). In exponential notation, this is represented as \(10^{6}\).

(f) k

The abbreviation "k" represents the prefix "kilo," which means a factor of one thousand or \(1,000\). In exponential notation, this is represented as \(10^{3}\).

(g) n

The abbreviation "n" represents the prefix "nano," which means a factor of one billionth or \(\frac{1}{1,000,000,000}\). In exponential notation, this is represented as \(10^{-9}\).

(h) m

The abbreviation "m" represents the prefix "milli," which means a factor of one-thousandth or \(\frac{1}{1,000}\). In exponential notation, this is represented as \(10^{-3}\).

(i) p

The abbreviation "p" represents the prefix "pico," which means a factor of one trillionth or \(\frac{1}{1,000,000,000,000}\). In exponential notation, this is represented as \(10^{-12}\).

Key concepts

Understanding Metric Prefixes

Metric prefixes are essential for expressing the magnitude of measurements in a clear, concise, and standardized manner. They help in quantifying the scale of units, which range from the very large to the very small. For example, the metric prefix 'kilo-' indicates a multiplication factor of a thousand, which is numerically represented in exponential notation as \(10^{3}\). Similarly, 'milli-', another commonly used metric prefix, specifies a thousandth, which in exponential notation is shown as \(10^{-3}\).

Here are a few metric prefixes explained with an attention to detail:

• Deci-: Signifies a tenth (\(10^{-1}\) or 0.1).
• Centi-: Indicates a hundredth (\(10^{-2}\) or 0.01).
• Micro-: Corresponds to a millionth (\(10^{-6}\) or 0.000001).
• Nano-: Represents a billionth (\(10^{-9}\) or 0.000000001).

Using these prefixes before a base unit of measurement not only simplifies the communication of large and small numbers but also aids in comprehending and visualizing them.

Exponential Notation Simplified

Exponential notation, a fundamental concept in dealing with numbers, is a way to express a number as a base raised to the power of an exponent. It is incredibly useful in representing very large or very small numbers succinctly. The basic form of exponential notation is \(a^n\), where 'a' is the base and 'n' is the exponent or power.

For example, the number 1,000 can be daunting to repeatedly write out in its full form. With exponential notation, it's simply written as \(10^3\). Similarly, one millionth, which is 0.000001, is more efficiently written as \(10^{-6}\). This style of notation is integral to fields like science and engineering, where precision is critical and numbers can be exceptionally large or small. It's also a key player in the metric system for representing the different scales of measurement.

Decoding Scientific Notation

Scientific notation is a powerful tool used to express very large or very small numbers in a more manageable form. It is closely related to exponential notation and is particularly useful in science and engineering. The typical format is \(m \times 10^n\), where 'm' is a number greater than or equal to 1 and less than 10, and 'n' is an integer. This approach allows for compact writing and easy comparison of magnitudes.

For instance, the speed of light is approximately 299,792,458 meters per second. In scientific notation, this speed is expressed as \(2.99792458 \times 10^8\) meters per second. When compared with another speed such as 340 meters per second (the speed of sound in air), which in scientific notation is \(3.4 \times 10^2\) meters per second, it's much easier to quickly infer that the speed of light is vastly greater.