# Scientific E Notation and conversion to decimal

I am working with satellite data and having fits understanding how to work with it because of the way the data is written (scientific E notation?). My math skills, unfortunately, are not where they should be for a programmer. I hope to find an answer here but I will give a small backstory to hopefully explain this better:

The data I am working on is lightning data, measured in units of 'Joules per Flash'. The US National Weather Service can display this data on their systems, and there is a legend that shows which color corresponds to the level of energy:

I managed to get ahold of the raw data, but it is in scientific notation and not on the 0-1500 scale as seen in the image. What is going on here?

For example, one value is $$3.657199E-16$$.

I figured I could convert that number to the nearest whole integer so I tried some converters online. (With my unfortunate math skills, this is what I resorted to).

I used this calculator and the result was a very tiny number in decimal form - $$0.0000000000000003657199$$

• any number out of the data values I have are like this. Does anyone have any idea, or could give me guidance on how to convert this number? I don't understand why this could be.

PS - If anyone finds this question to be unfitting or unsuitable to ask here, please tell me in the comments rather than downvote. I will happily delete the question if needed rather than lose reputation, I am just not sure where to ask but first, it would be helpful to know if the problem is math-related before moving on to other things.

• The very small number you got is right - this must be one of the places on the map that is black. Dec 22 '19 at 9:54
• Unfortunately it is not - the black areas are no-data values and they are all "0". Perhaps my issue is something else then. Thank you for looking into this. Dec 22 '19 at 9:56
• It looks like I needed to convert from joules to femtojoules! Thank you for your assistance again, it helped me be confident in what I was looking at. Dec 22 '19 at 10:04
• @David If your real objective was converting from J to fJ, that wasn't clear.
– J.G.
Dec 22 '19 at 10:11
• @J.G. Yes, it was not clear because I did not know it was being converted to fJ. This is cutting edge scientific data off of NOAA's latest satellites, and sadly not all things in life are not as clear as I would desire. Life can be complicated. Dec 22 '19 at 10:14

## 2 Answers

The conversion is very simple: simply shift the decimal dot 16 places to the left. Complete with zeroes if necessary. The meaning of the notation is $$$$3.657199\times 10^{-16} = \frac{3.657199}{10^{16}}$$$$ hence $$3.657199$$ divided by 10 million billions.

Of course with $$6.02E23$$ you would shift the dot 23 places to the right!

Edit I want to add that a value such as 3.65E-16 in numerical data may sometimes simply mean zero if it is the result of a computation in typical 64 bits floating precision. For example on my computer, if I compute

>>> 3 * 0.1 - 0.3
5.551115123125783e-17


This value simply means an error due to the limited precision of the computation. It doesn't have another meaning, let it be Joules or femto Joules!

• My problem was that I needed to convert joules to femtojoules, however your answer helped me realize I was on the right track. Thank you. Dec 22 '19 at 10:05

Scientific E notation is used in displaying big or small numbers with some characters. 'E' notation is used in raw data, calculators, computers, etc. But mathematic notation is widely used by mathematicians. Let me show first the math notation:

Mathematicians write such big or small numbers on this way:

$$3000000000 = 3 × 10^9$$ $$0.000000003 = 3 × 10^{-9}$$

Or simply: If you have number $$x$$ and $$n$$ zeroes, then written number is $$\overline{x000...000} = x × 10^n$$

(Notation $$n=\overline{xxxxx}$$ means that these are the digits of the number n. For example, if $$x=1$$, then $$n=\overline{xxxxx}=11111$$)

In another direction: If you have $$n$$ zeroes and number $$x$$ (including this before coma), then the number is written as $$\overline{0.000...0000x} = x * 10 ^ {-n}$$

But scientific E notation writes numbers on this way: $$n × 10 ^ {x} = nEx$$ because computers can't easily store the data in exponents, but with 'E' is everything easier.

You should know also what are the metric prefixes.

The definition of the metric prefix is:

A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. While all metric prefixes in common use today are decadic, historically there have been a number of binary metric prefixes as well. (Wikipedia)

Prefixes are defined nowadays in the decadic system. In fact, you should keep in mind, that k in kg means $$×10^3$$, so: $$1 kg = 1 × 10^3 g$$

Every prefix means the exponent of ten. You will convert it in the best way if you will use femto. So look at the table at Wikipedia and you will find the prefix femto at the value $$×10^{-15}$$, so we can write the result as: $$3.657199E-16J=3.657199 × 10^{-16}J=3.657199 × 10 ^ {-1} fJ = 3.657199 × 0.1 fJ = 0.3657199 fJ$$

The prefix femto (f) destroyed $$×10^{-15}$$ in the second step (when we got $$×10^{-1}$$), so it stays only $$×10^{-1}$$, which is equal to $$×1/10^1=×1/10=0.1$$

Hope that this helped and if you still have any question, ask me in the comments.

P.S.
Thank you for the awesome question. Now you will know a lot about the metric prefixes. And sorry for the extra long question.

• I had already accepted the other answer, but your effort to help me understand is well appreciated! This is going to be useful for me. long live SO Dec 25 '19 at 14:11
• @David Thank you! Dec 26 '19 at 17:56