System of Units & Power of 10

Demonstrative Video

System of Units

  • In the past, most commonly used were the English and Metric

  • English system is based on a single standard, the metric is subdivided into two interrelated standards: the MKS and the CGS

  • MKS system uses Meters, Kilograms, and Seconds, while the CGS system uses Centimeters, Grams, and Seconds.

  • Need for a standard set of units to be adopted by all nations has become increasingly obvious.

  • In 1960, the General Conference held by International Bureau of Weights and Measures located at France, adopted a system called Le Système International d’Unités (International System of Units), which has the international abbreviation SI.

Comparison of system of Units

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Physical quantity Symbol Unit Symbol Measuring device
Current I Ampere A Amperemeter
Voltage U Volt V Voltmeter
Power P Watt W Powermeter
Resistance R Ohm \(\Omega\) Ohmmeter
Capacitance C Farad F Capacitance meter
Inductance L Henry H Inductance meter
Frequency f Hertz Hz Oscilloscope
Period T Second S Oscilloscope
Charge Q Coulomb C Charge meter
Conductance G Siemens S Conductivity meter

Name Symbol Base equivalent
Volt V \(kg \cdot m^2 \cdot s^{-3} \cdot A^{-1}\)
Watt W \(kg \cdot m^2 \cdot s^{-3}\)
Ohm \(\Omega\) \(kg \cdot m^2 \cdot s^{-3} \cdot A^{-2}\)
Farad F \(kg^{-1} \cdot m^{-2} \cdot s^{4} \cdot A^{2}\)
Henry H \(kg \cdot m^2 \cdot s^{-2} \cdot A^{-2}\)
Hertz Hz \(s^{-1}\)
Coulomb C \(A \cdot s\)
Siemens S \(kg^{-1} \cdot m^{-2}\cdot s^{3} \cdot A^{2}\)

Comparison of various Units

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Power of 10

  • Relative magnitude of the various units of measurement very large and very small numbers will frequently be encountered

  • To ease the difficulty of mathematical operations with numbers of such varying size, powers of ten are usually employed

  • \(10000 = 10^{+4}~\Rightarrow\) count number of zeros

  • \(0.00001 = 10^{-5}~\Rightarrow\) zeros+1

  • \(10^{n} = \dfrac{1}{10^{-n}}\)

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Basic Arithmetic Operation of Power of 10

  • \[A \times 10^{n} \pm B \times 10^{n} = \left(A\pm B\right) \times 10^{n}\]
    Addition and Subtraction
  • \[\left(A\times 10^{n}\right)\left(B\times 10^{m}\right) = (A)(B)\times 10^{n+m}\]
    Multiplication
  • \[\dfrac{A\times 10^{n}}{B\times 10^{m}} = \dfrac{A}{B} \times 10^{n-m}\]
    Division
  • \[\left(A\times 10^{n}\right)^{m} = A^{m} \times 10^{nm}\]
    Powers

Fixed-Point, Floating-Point, Scientific, & Engineering Notation

  • If powers of ten are not employed, they are written in the fixed-point or floating-point notation.

  • The fixed-point format requires that the decimal point appear in the same place each time.

  • In the floating-point format, the decimal point will appear in a location defined by the number to be displayed.

  • \[\frac{1}{3}=0.333 \quad \frac{1}{16}=0.063 \quad \frac{2300}{2}=1150.000\]
    Fixed-point with thousandths place accuracy:
  • \[\frac{1}{3}=0.333333333333 \quad \frac{1}{16}=0.0625 \quad \frac{2300}{2}=1150\]
    Floating point format:

  • Scientific (also called standard) notation and engineering notation make use of powers of ten with restrictions on the mantissa (multiplier) or scale factor (power of the power of ten).

  • Scientific notation requires that the decimal point appear directly after the first digit greater than or equal to 1 but less than 10.

  • \[\frac{1}{3}=3.33333333333 \mathrm{E}-1 \quad \frac{1}{16}=6.25 \mathrm{E}-2 \quad \frac{2300}{2}=1.15 \mathrm{E} 3\]
    Within the scientific notation, the fixed- or floating-point format can be chosen.
  • \[\frac{1}{3}=333.333333333 \mathrm{E}-3 \quad \frac{1}{16}=62.5 \mathrm{E}-3 \quad \frac{2300}{2}=1.15 \mathrm{E} 3\]
    Engineering notation specifies that all powers of ten must be multiples of 3, and the mantissa must be greater than or equal to 1 but less than 1000

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