Two students of \(60\,\text{kg}\) sit on a bench in a school corridor, their centres \(1\,\text{m}\) apart. The law of gravitation says that they pull on each other with a force of \(0.000\,000\,24\,\text{N}\). A hydrogen atom has a radius of \(0.000\,000\,000\,053\,\text{m}\). The Earth has a radius of \(6\,400\,000\,\text{m}\). All three numbers appear in this book, and each one is hard to read.
A number and a power of ten
Physics works with very large and very small numbers. Count the zeros in the force between the students, and it is easy to lose one. A lost zero is an error by a factor of ten. So physicists write every such number in a short standard form, as a number between 1 and 10 times a power of ten.
A power of ten is ten multiplied by itself a number of times. \(10^3 = 10 \times 10 \times 10 = 1000\), and \(10^6\) is a million. A negative power means division: \(10^{-1} = 0.1\) and \(10^{-3} = 0.001\). Finally, \(10^0 = 1\). The small raised number is the exponent, and it counts the steps of ten.
To write a number in scientific notation, move the decimal point until one non-zero digit stands before it. Then count the places moved. For the Earth, \(6\,400\,000.\) becomes \(6.4\) after the point moves six places to the left, so the radius is \(6.4 \times 10^6\,\text{m}\). For the atom, the point moves eleven places to the right, so the radius is \(5.3 \times 10^{-11}\,\text{m}\).
A number is in scientific notation when it is written as \(a \times 10^b\). The number \(a\) is at least 1 and less than 10, and the exponent \(b\) is a whole number, positive, negative or zero. The digits of \(a\) carry the measured value; the power of ten only places the decimal point.
The yellow light of a sodium street lamp has a wavelength of \(0.000\,000\,589\,\text{m}\). How is this written in scientific notation?
Multiplying by adding exponents
Multiply \(10^3\) by \(10^2\). The first is three tens multiplied together and the second is two more, so the product has five tens: \(10^3 \times 10^2 = 10^5\). To multiply powers of ten, add their exponents. Division removes tens, so \(10^5/10^2 = 10^3\). To divide, subtract the exponents. The same two rules hold for negative exponents.
The division rule also explains the meaning of a zero or a negative exponent. The derivation below uses nothing else.
\[ \dfrac{10^m}{10^n} = 10^{m - n} \]
Why. dividing by \(10^n\) removes \(n\) of the \(m\) tens in the numerator.
\[ \dfrac{10^n}{10^n} = 10^{0} \]
Why. put \(m = n\) in the rule of step 1.
\[ 10^0 = 1 \]
Why. any number divided by itself is 1, so the rule stays true only if \(10^0 = 1\).
\[ \dfrac{10^0}{10^n} = 10^{-n} \]
Why. put \(m = 0\) in the rule of step 1.
\[ 10^{-n} = \dfrac{1}{10^n} \]
Read it. replace \(10^0\) by 1, from step 3. A negative power of ten is one over the positive power.
A number in scientific notation has two parts, so multiply the parts separately. Multiply the coefficients, and add the exponents. If the new coefficient reaches 10 or more, move the point once more. For example, \((6 \times 10^4) \times (3 \times 10^{-7}) = 18 \times 10^{-3}\), and \(18 \times 10^{-3} = 1.8 \times 10^{-2}\). Squaring follows the same rule. The radius of the hydrogen atom gives \((5.3 \times 10^{-11})^2 = 28.09 \times 10^{-22} = 2.81 \times 10^{-21}\).
Two students of mass \(60\,\text{kg}\) each sit on a bench with their centres \(1\,\text{m}\) apart. Find (a) the gravitational force \(F = Gm_1m_2/r^2\) between them, with \(G = 6.67 \times 10^{-11}\,\text{N m}^2/\text{kg}^2\). Then (b) compare the force with the weight of one student, taking \(g = 9.8\,\text{m/s}^2\).
(a) First the masses: \(m_1m_2 = 60 \times 60 = 3600 = 3.6 \times 10^3\,\text{kg}^2\), and \(r^2 = 1\,\text{m}^2\).
Then \(F = 6.67 \times 10^{-11} \times 3.6 \times 10^3\). Multiply the coefficients, \(6.67 \times 3.6 = 24.0\), and add the exponents, \(-11 + 3 = -8\).
So \(F = 24.0 \times 10^{-8}\,\text{N} = 2.4 \times 10^{-7}\,\text{N}\), the \(0.000\,000\,24\,\text{N}\) of the bench.
(b) The weight is \(60 \times 9.8 = 588\,\text{N} = 5.88 \times 10^2\,\text{N}\). Divide: \(5.88/2.4 = 2.45\), and \(2 - (-7) = 9\).
The weight is \(2.45 \times 10^9\) times the pull, about two and a half billion times. This is why nobody feels the pull of a neighbour.
What is \((4 \times 10^5) \times (5 \times 10^{-8})\) in scientific notation?
In a hydrogen atom the electron is \(5.3 \times 10^{-11}\,\text{m}\) from the proton. Coulomb's law gives the electric force as \(F = ke^2/r^2\), with \(k = 9 \times 10^9\) and \(e = 1.6 \times 10^{-19}\), in SI units. Write \(F\) as \(a \times 10^{-8}\,\text{N}\) and enter \(a\).
How big, to the nearest power of ten
Often we need only a rough idea of the size of a quantity. Is it nearer a millimetre or a kilometre? The order of magnitude answers this question. It is the power of ten nearest to the quantity. Two quantities of the same order differ by less than a factor of about ten.
Fig 1.1 puts eight lengths from this book on one ruler. Each step to the right multiplies the length by ten, so a nucleus and the Earth fit on one line. Choose a chip and read the length in both forms. The decimal form grows long at both ends of the ruler; the scientific form never does.
To find the order of magnitude, write the quantity as \(a \times 10^b\). If \(a\) is 5 or less, the order is \(b\). If \(a\) is more than 5, the quantity is nearer the next power of ten, and the order is \(b + 1\). Fig 1.2 shows the rule for the Moon and the Earth.
A few orders from the book show the range. A student is \(1.7\,\text{m}\) tall, of order 0. The hydrogen atom, \(5.3 \times 10^{-11}\,\text{m}\), is of order \(-10\), because \(5.3\) is more than 5. The density of a nucleus is \(2.3 \times 10^{17}\,\text{kg/m}^3\), of order 17. Water has \(1.0 \times 10^3\,\text{kg/m}^3\), of order 3. So nuclear matter is about fourteen orders of magnitude denser than water.
Avogadro's number is \(6.02 \times 10^{23}\). What is its order of magnitude?
Nuclear matter has a density of \(2.3 \times 10^{17}\,\text{kg/m}^3\), and water has \(1.0 \times 10^3\,\text{kg/m}^3\). About how many orders of magnitude denser than water is nuclear matter?
The order of magnitude is a rough label, and its boundary at 5 is a convention. A radius of \(4.9 \times 10^6\,\text{m}\) is of order 6 and one of \(5.1 \times 10^6\,\text{m}\) is of order 7, yet the two differ by only 4 %. Some books place the boundary at \(\sqrt{10} \approx 3.16\) instead. So never use orders alone to decide which of two close quantities is larger. Compare the numbers themselves.
The phenomenon is now answered. The pull between the students is \(2.4 \times 10^{-7}\,\text{N}\), of order \(-7\). The atom is \(5.3 \times 10^{-11}\,\text{m}\) and the Earth \(6.4 \times 10^6\,\text{m}\). Each number is now one short line, and the arithmetic is done on the exponents.
The Greeks of Archimedes' time had no name for a number larger than a myriad, ten thousand. Many people believed that the grains of sand could never be counted. Archimedes answered them in a short book, The Sand Reckoner. He built names for larger numbers, each one a myriad of myriads of the one before. Then he estimated how many grains of sand would fill the whole universe as the astronomers of his day imagined it. His answer was fewer than \(10^{63}\) in our notation. The idea of counting by powers is his.
Scientific notation is used on nearly every page of this book. Gravitation begins with \(G = 6.67 \times 10^{-11}\,\text{N m}^2/\text{kg}^2\) and the mass of the Earth. Electrostatics compares the electric and the gravitational pulls in an atom, a ratio of \(2.3 \times 10^{39}\). Current electricity counts \(8.5 \times 10^{28}\) free electrons in a cubic metre of copper. The nucleus chapter finds the density of nuclear matter, and the kinetic theory of gases counts molecules with Avogadro's number.
- Writing \(589 \times 10^{-9}\) or \(0.589 \times 10^{-6}\) as the scientific form.The coefficient must lie between 1 and 10. Move the point and change the exponent: \(5.89 \times 10^{-7}\).
- Multiplying the exponents when multiplying two numbers.Multiply the coefficients and add the exponents: \(10^4 \times 10^{-7} = 10^{-3}\).
- Forgetting to square the power of ten when squaring a number.\((5.3 \times 10^{-11})^2 = 28.09 \times 10^{-22}\). Both the coefficient and the power are squared.
- Taking the order of magnitude of \(6.02 \times 10^{23}\) as 23.When the coefficient is more than 5, the order is one more than the exponent: 24.