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More specifically, my question is why is the point always after the first digit? Why have (for example), 7.5 x 1018 as opposed to 75 x 1017? The former makes it easier to read (when you literally want to say the number of zeros) in cases where n is a negative integer, but harder when it is positive.

The possibility I can think of is consistency. Having the separation always in the same spot could possibly avoid a lot of mistakes, but so could never using it at all. Perhaps having a negative n is so common that it’s worth it, or it’s done for historical reasons, but these are all hypothesis I can come up with, I’d like to know for sure.

user137369
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    I see basically two reasons: (1) you immediately see the order of magnitude (it's easier if you have, say $ 1.003\cdot 10^{8} $, rather than $ 1003\cdot 10^{5} $), and the order of magnitude is far more interesting than the number of actual zeroes. (2) It is also useful to specify the accuracy whereby the number is known. Writing 100, for instance, would lead to ambiguity for the number of known digits, while writing $ 1.00 \cdot 10^2 $ would not. – derpy Apr 29 '14 at 16:14

3 Answers3

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Your suggestion would make comparisons more difficult: What is bigger? $75\cdot 10^{17}$ or $74999\cdot 10^{15}$? Also, it is easier to round: Give an rough approximation for $1993\cdot 10^8$ times $31\cdot 10^7$ is hard to do. But to give a rough approximation to $1.993\cdot 10^{11}$ times $3.1\cdot 10^{8}$ is easy - that should be about $6\cdot10^{19}$. Also, I don't see any advantage in being able to spell "seven, five, and 17 zeroes" or "seven, four, nine, nine, nine, and then 15 zeroes".

Apart from that, there are in fact several standards: One is to use a mantissa that is always between $0.1$ and $1$; another to use a mantissa that is always between $1$ and $10$; and finally to use a mantissa between $1$ and $1000$ and to have an exponent a multiple of $3$. The latter method also is meant to facilitate speaking numbers in terms of -illions, or to translate to SI prefixs such as nano, mikro, milli, kilo, mega, giga.

Hagen von Eitzen
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  • Upvoted the answer as I can only accept one. I chose Joel’s answer for the historic context, but thank you for taking the time on this. – user137369 Apr 29 '14 at 17:33
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The big reason why you want to use scientific notation is to separate magnitude and the value. If you want to tell which of the numbers $5.8\times 10^{2}$, $6.2\times 10^3$ and $2.0\times 10^{2}$ is bigger then it is clear that $6.2\times 10^3 = 6,200$ is larger than the rest.

Another reason is that computation used to be done with slide rules. These simple calculators could multiply any two numbers less than 10. Thus if you want to multiply: $$5.8\times 10^{2} \text{ and } 6.2\times 10^3$$

Then you use the slide rule to compute $5.8 \cdot 6.2 = 35.96$, and then put the tens back in to get $$35.96\times 10^5 = 3.596 \times 10^6$$

Joel
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  • Thank you. This was asked purely in the name of curiosity (been wondering it for years). I like both your answer and Hagen’s, but I’m accepting this one due to the inclusion of the historic context. – user137369 Apr 29 '14 at 17:29
  • I am glad we could answer your question. – Joel Apr 29 '14 at 17:52
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Before calculators existed, logarithms were computed from tables. those tables listed the (base 10) logarithms of numbers $1 \le x \lt 10$. Thus, for example the log of $2$ is $0.3010$. So log $2 \times 10^5 = 5.3010$.

Steven Alexis Gregory
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