Question

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

Short answer

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

Step-by-step solution

(a) d - Deci

The abbreviation "d" represents "deci" which is 1/10 or 10⁻¹ times the base unit.

(b) c - Centi

The abbreviation "c" represents "centi" which is 1/100 or 10⁻² times the base unit.

(c) f - Femto

The abbreviation "f" represents "femto" which is 1/1,000,000,000,000,000 or 10⁻¹⁵ times the base unit.

(d) μ - Micro

The abbreviation "μ" represents "micro" which is 1/1,000,000 or 10⁻⁶ times the base unit.

(e) M - Mega

The abbreviation "M" represents "mega" which is 1,000,000 or 10⁶ times the base unit.

(f) k - Kilo

The abbreviation "k" represents "kilo" which is 1,000 or 10³ times the base unit.

(g) n - Nano

The abbreviation "n" represents "nano" which is 1/1,000,000,000 or 10⁻⁹ times the base unit.

(h) m - Milli

The abbreviation "m" represents "milli" which is 1/1,000 or 10⁻³ times the base unit.

(i) p - Pico

The abbreviation "p" represents "pico" which is 1/1,000,000,000,000 or 10⁻¹² times the base unit.

Key concepts

Metric Prefixes

Metric prefixes are shorthand symbols used in science and mathematics to easily express different orders of magnitude and quantities. They make it simpler to handle very large or very small numbers by replacing them with a prefix attached to a unit. This practice saves time and simplifies calculations.

• Examples:
• Deci (d): Represents 10^-1, or one-tenth of a unit.
• Centi (c): Represents 10^-2, or one-hundredth of a unit.
• Milli (m): Represents 10^-3, or one-thousandth of a unit.
• Micro (μ): Represents 10^-6, or one-millionth of a unit. Latin letter "mu" is used for this prefix.
• Nano (n): Represents 10^-9, or one-billionth of a unit.
• Pico (p): Represents 10^-12, or one-trillionth of a unit.
• Femto (f): Represents 10^-15, or one-quadrillionth of a unit.
• Kilo (k): Represents 10³, or one thousand times the base unit.
• Mega (M): Represents 10⁶, or one million times the base unit.

Metric prefixes originate from the metric system, which is based on powers of ten to ensure utmost convenience in expressing different magnitudes. This system is utilized globally, making scientific communication easier and more standardized.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to conveniently write in decimal form. It makes use of powers of ten to represent values, simplifying arithmetic operations and comparisons between numbers of different magnitudes.
Imagine you want to write the distance from the Earth to the Sun, approximately 149,600,000 kilometers. In scientific notation, this is written as 1.496 x 10⁸ kilometers. Here's how it works:

• The number consists of two parts: a coefficient and an exponent.
• The coefficient (1.496 in the example) is a compact form with a maximum of three digits before the decimal point.
• The exponent indicates how many places the decimal point must be moved to convert back to the standard numeric form. If positive, move the point to the right; if negative, move it to the left.

Scientific notation is especially useful in scientific fields where very precise calculations are frequent, such as physics, chemistry, and astronomy. It is intuitive and efficient at handling extremely large or small values which can often have many trailing or leading zeroes.

Powers of Ten

The concept of powers of ten is foundational in understanding both metric prefixes and scientific notation. It involves using the number ten raised to various exponents to scale numbers either up or down.

• Positive powers: When ten is raised to a positive integer, like 10³, it represents 1,000. This scales numbers up, giving a sense of 'greater' quantity.
• Negative powers: A negative exponent indicates division by ten multiple times. For instance, 10^-3 is 0.001, effectively scaling numbers down to show 'smaller' quantities.

Understanding powers of ten is crucial when working with very large and very small numbers. For example, writing 10⁶ (meaning one million) is much more efficient than writing out a million zeros. Powers of ten simplify mathematical operations such as multiplication and division, by allowing you to 'add' or 'subtract' exponents.