Question

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

Short answer

The exponential notations for the given abbreviations are: (a) d: \(10^{-1}\) (b) c: \(10^{-2}\) (c) f: \(10^{-15}\) (d) µ: \(10^{-6}\) (e) M: \(10^{6}\) (f) k: \(10^{3}\) (g) n: \(10^{-9}\) (h) m: \(10^{-3}\) (i) p: \(10^{-12}\)

Step-by-step solution

(a) d

The abbreviation 'd' stands for deci, which means one-tenth of a unit. In exponential notation, this is represented as \(10^{-1}\).

(b) c

The abbreviation 'c' represents centi, which means one-hundredth of a unit. In exponential notation, this is represented as \(10^{-2}\).

(c) f

The abbreviation 'f' indicates femto, which represents one trillionth of a unit. In exponential notation, this is represented as \(10^{-15}\).

(d) µ

The abbreviation 'µ' stands for micro, which means one millionth of a unit. In exponential notation, this is represented as \(10^{-6}\).

(e) M

The abbreviation 'M' represents mega, which indicates one million of a unit. In exponential notation, this is represented as \(10^{6}\).

(f) k

The abbreviation 'k' indicates kilo, which means one thousand of a unit. In exponential notation, this is represented as \(10^{3}\).

(g) n

The abbreviation 'n' stands for nano, which means one billionth of a unit. In exponential notation, this is represented as \(10^{-9}\).

(h) m

The abbreviation 'm' represents milli, which means one-thousandth of a unit. In exponential notation, this is represented as \(10^{-3}\).

(i) p

The abbreviation 'p' stands for pico, which indicates one trillionth of a unit. In exponential notation, this is represented as \(10^{-12}\). Here is the summary of the given abbreviations and their exponential notations: (a) d: \(10^{-1}\) (b) c: \(10^{-2}\) (c) f: \(10^{-15}\) (d) µ: \(10^{-6}\) (e) M: \(10^{6}\) (f) k: \(10^{3}\) (g) n: \(10^{-9}\) (h) m: \(10^{-3}\) (i) p: \(10^{-12}\)

Key concepts

Exponential Notation

Exponential notation is a mathematical method used to represent numbers in a concise format by indicating how many times a number should be multiplied by itself. This notation helps in simplifying complex calculations and dealing with extremely large or small numbers. For example, the number 1,000 can be written in exponential notation as \(10^3\), where the exponent '3' indicates that the base number 10 is multiplied by itself three times.

This concept is particularly helpful when working with measurements in physics, chemistry, and engineering, where a vast range of values may exist. Key points to remember:

• A positive exponent indicates a multiplication of the base number (e.g., \(10^6 = 1,000,000\)).
• A negative exponent signifies a division or one over the base raised to the positive value of the exponent (e.g., \(10^{-3} = 0.001\)).
• Exponential notation is frequently used to express SI prefixes, which are part of the metric system.

Metric System

The metric system is an international standard of measurement that is based on the powers of ten. It uses a set of base units and prefixes, such as meter, gram, and liter, to measure length, mass, and volume, respectively. The metric system makes it easier to understand and convert between different units because each prefix corresponds to a specific power of ten.

Let's explore some common metric system prefixes and their meanings:

• Kilo (k): Represents \(10^3\), or one thousand units.
• Mega (M): Represents \(10^6\), or one million units.
• Milli (m): Represents \(10^{-3}\), or one-thousandth of a unit.
• Micro (µ): Represents \(10^{-6}\), or one-millionth of a unit.
• Nano (n): Represents \(10^{-9}\), or one-billionth of a unit.

This system simplifies calculations and provides a universal standard, making scientific communication across different countries much more consistent and reliable.

Scientific Notation

Scientific notation is a method to easily write very large or very small numbers in a compact form. It helps scientists, engineers, and students to work with these numbers without needing to write out many zeros, which significantly reduces the risk of error.

In scientific notation, numbers are expressed in the form \(a \times 10^b\), where 'a' is a number greater than or equal to 1 but less than 10, and 'b' is an integer that shows the number of places the decimal point has been moved.

• For example, the number 5,000 can be written as \(5 \times 10^3\).
• Conversely, \(0.005\) can be expressed as \(5 \times 10^{-3}\).
• This notation allows for ease of multiplication and division of expressed numbers, as you only need to work with the coefficients 'a' and add or subtract exponents 'b'.

The use of scientific notation is widespread in scientific disciplines as it facilitates the handling of data and calculation, ensuring precise communication of significant figures.