Why is scientific notation of the form a.bcd×10^n?

Question

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.

Answer 1

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 $$35.96\times 10^5 = 3.596 \times 10^6$$

Answer 2

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.

Answer 3

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 $2 \times 10^5 = 5.3010$.