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Standard index form of numbers
Calculations are often performed with the help of powers in natural sciences, since it is easier to express very large and very small numbers with the help of powers.
For example the propagation speed of light is \latex{300,000,000} \latex{\frac{m}{s}}, expressed in a simpler form: \latex{3\times 10^{8} } \latex{\frac{m}{s}}. The coefficient of linear thermal expansion of aluminium is \latex{0.000024 \frac{1}{°C} }, shortly \latex{2.4\times 10^{-5} \frac{1}{°C}}.
The numbers can be expressed in several ways in product form; it is practical to select the form where the power of \latex{ 10 } with integer index is multiplied by a number between \latex{ 1 } and \latex{ 10 }.
DEFINITION: In the case of a real number the form \latex{a\times10^{k}} is called the standard index form (in scientific notation) of the number, \latex{1\leq \mid a\mid \lt 10} and \latex{k\in \Z}.
For example:
10,000 = 104;
321,000,000 = 3.21 × 108;
0.00000000068 = 6.8 × 10–10.
321,000,000 = 3.21 × 108;
0.00000000068 = 6.8 × 10–10.
Example 1
How long is a \latex{ light-year? }
Solution
The \latex{ light-year } is the distance covered by the light in one \latex{ year }. The speed of light is \latex{c = 3\times 10^{8}\,\frac{m}{s}}, the time elapsed when considering a \latex{ year } with \latex{ 365 } \latex{ days } is
\latex{t = 365 \times 24 \times 60 \times 60 } \latex{s}, when rounded: \latex{3.15 \times 10^{7} s}.
The distance is:
\latex{3 \times 10^{8} \times 3.15 \times 10^{7} \,m = 9.45 \times 10^{15}\, m = 9.45 \times 10^{12}} \latex{km}.
To demonstrate how long this distance is let us compare it to the distance between the Sun and the Earth, which is approximately \latex{1.5 \times 10^{8}} \latex{km}.
Their quotient is:
\latex{\frac{9.45\times 10^{12}}{1.5\times 10^{8}} =6.3\times 10^{4}},
so a \latex{ light-year } is more than \latex{ 60,000 } times the distance between the Earth and the Sun.
Let us represent the distance between the Earth and the Sun. The speed
of a racing car can be \latex{ 300 } \latex{ \frac{km}{h} } . With this speed
of a racing car can be \latex{ 300 } \latex{ \frac{km}{h} } . With this speed
\latex{\frac{1.5\times 10^{8}\,km}{300\,\frac{km}{h} } = \frac{1.5\times 10^{8}}{3\times 10^{2}}\,h = 0.5\times10^{6}\,h = 5\times 10^{5}\,h \approx 57 \; years}
would be needed to cover the distance between the Earth and the Sun.
Example 2
What is the total mass of the Moon and the Earth?
Solution
The mass of the Earth is \latex{5.974 \times 10^{24}\; kg}, and the mass of the Moon is \latex{7.347 \times 10^{22} \;kg}.
The total mass is:
\latex{5.974\times 10^{24}+7.347\times10^{22}\,kg=597.4\times10^{22}+7.347\times10^{22}\,kg=}\latex{=604.747\times10^{22}\,kg=6.048\times10^{24}\,kg.}
The mass of the Sun and the
planets:
planets:
Sun
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
\latex{ 1.990\times 10^{30} } \latex{ kg }
\latex{ 3.286\times 10^{23} } \latex{ kg }
\latex{ 4.896\times 10^{24} } \latex{ kg }
\latex{ 5.974\times 10^{24} } \latex{ kg }
\latex{ 6.392\times 10^{23} } \latex{ kg }
\latex{ 1.899\times 10^{27} } \latex{ kg }
\latex{ 5.684\times 10^{26} } \latex{ kg }
\latex{ 8.698\times 10^{25} } \latex{ kg }
\latex{ 1.028\times 10^{26} } \latex{ kg }
Example 3
The mass of a proton is \latex{m_{p}=1.67\times 10^{-27}\,kg}. How many times greater is this mass than the \latex{m_{e}=9.1\times10^{-31}\,kg} mass of an electron?
Solution
Their quotient is
\latex{\frac{1.67\times10^{-27}}{9.1\times10^{-31}} \approx 0.18\times10^{4}=1,800}.
Thus the mass of a proton is about \latex{ 1,800 } times the mass of an electron.
Example 4
The mass of one mole of helium is \latex{4\,g}, which means that there are 6 × 1023 atoms in \latex{ 4 } \latex{g} of helium. What is the mass of one helium atom?
Solution
The missing mass is
\latex{\frac{4}{6\times10^{23}}\,g\approx 0.6667\times10^{-23}\,g= 6.667\times 10^{-24}\,g}
Example 5
Which distance is longer: \latex{48,000,000\,nanometres} or \latex{ 50,000 } \latex{micrometres?}
Solution
Let us express both distances in \latex{ metres }.
48,000,000 \latex{nanometres} = (4.8 × 107) × 10–9 \latex{m} = 4.8 × 10–2 \latex{m};
50,000 \latex{micrometres} = (5 × 104) × 10–6 \latex{m} = 5 × 10–2 \latex{m}.
50,000 \latex{micrometres} = (5 × 104) × 10–6 \latex{m} = 5 × 10–2 \latex{m}.
The first distance is \latex{ 4.8 } \latex{cm}, the second one is \latex{ 5 } \latex{cm}, and thus \latex{ 50,000 } \latex{micrometres} is the longer distance.
Exercises
{{exercise_number}}. If the mass of one grain of wheat is 5 × 10–2 \latex{grams}, then how many grains are there in 1\latex{ton} of wheat?
{{exercise_number}}. How much time does the light of the Sun need to reach the Earth?
{{exercise_number}}. The charge of the electron is 1.6 × 10–19 \latex{coulombs}. How many electrons are needed to transfer 0.001 \latex{coulombs} of charge?
{{exercise_number}}. Our heart beats about 70 times \latex{ per } \latex{ minute. } Calculate how many times your heart has beaten so far.
{{exercise_number}}. Calculate your body height in \latex{ light-years. }
{{exercise_number}}. What percentage of the mass of the Sun is the mass of the planets of the Solar System? (Data can be found above.)
Puzzle
Give numbers \latex{a}, \latex{b}, \latex{c}, \latex{d} so that \latex{a} × 10–7 + \latex{b} × 10–5 + \latex{c} × 10–6 + \latex{d} × 10–8 = 1.05 × 10–6.
