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# Scientific notation

There are many small and large numbers to handle in this world. Such as the mass of a single water molecule1 and the mass of the Moon:2

$$m_\text{water}= 0.000\,000\,000\,000\,000\,000\,000\,000\,03\,\mathrm{kg}$$

$$m_\text{moon}= 70\,000\,000 \,000 \,000 \,000 \,000 \,000 \,\mathrm{kg}$$

Too many digits to handle. Let’s write them with prefixes:

$$m_\text{water}= 300\,\mathrm{zg}$$

$$m_\text{moon}= 70 \,\mathrm{Yg}$$

Easier to handle, but harder to compare. Can’t we invent some kind of writing style – a notation – that makes them more comparable? Actually, why don’t we just compare zeroes!

• $m_\text{moon}$ has 22 zeroes after the first digit.
• $m_\text{water}$ has 26 zeroes before the first digit.

Each zero on the right corresponds to ‘multiplied by 10’. Each zero on the left corresponds to ‘divided by 10’. So, $7$ is multiplied with ten 22 times, meaning multiplied with $10^{22}$, and 3 is divided with ten 26 times, meaning divided with $10^{26}$, which with a negative exponent means multiplying with $10^{-26}$. All in all, let’s write the values as

$$m_\text{water}= 3\times 10^{-26}\,\mathrm{kg}$$

$$m_\text{moon}= 7 \times 10^{22} \,\mathrm{kg}$$

We could have used the dot $\cdot$ symbol for multiplication. But, let’s use the $\times$ symbol to remind us that this is a rewriting of one number rather than two separate numbers multiplied. Doesn’t matter, it still means ‘multiply’.

It is now very comparable. The exponents on the 10s are quite influential with large control over the magnitude of the number; we could say that they tell the number’s order of magnitude.[3,4] Saying ‘our profits rose 3 orders of magnitude‘ may seem fairly subtle but it actually means an increase of one thousand times the value!

Also, we can do much easier math with numbers written in this way, because we now can use the exponent rules of algebra

This is universally called scientific notation and is a standardised way to show values in scientific papers and in technical work.[4] The procedure is simple:3

1. Place the decimal point after the first digit.
2. Then multiply with magnitudes of 10 to make it fit.4

References:

1. Avogadro's Number Example Chemistry Problem’ (web page), PhD Anne Marie Helmenstine, ThoughtCo, 2020, www.thoughtco.com/avogadros-number-example-chemistry-problem-609541 (accessed Jan. 28st, 2020)
2. Mass of the Moon’ (web page), Fraser Cain, Universe Today, 2008, www.universetoday.com/19728/mass-of-the-moon (accessed Jan. 28st, 2020)
3. Common errors in English usage’ (book), Paul Brians, William, James & Co., 3rd ed., 2013, brians.wsu.edu/2016/05/19/orders-of-magnitude, ISBN 9781590282632
4. Significant Figures and Order of Magnitude’ (web page), Lumen Boundless Physics, 2017, courses.lumenlearning.com/boundless-physics/chapter/significant-figures-and-order-of-magnitude (accessed Mar. 12th, 2020)