Scientific Notation Calculators

Scientific notation is a method of expressing numbers commonly used by scientific notation calculators and also by humans. The scientific notation style includes exponents and provides a way to write numbers that are too big or too small to write in traditional styles. Engineering fields commonly use this notation in order to simplify equations and formulas.

For example, 1 trillion dollars would be written in a traditional style like this:

1000000000000

That is '1' followed by 12 zeroes. The number poses problems because it's altogether too easy for human beings to miscount or miswrite the string of zeroes, thereby introducing error into the expression.


Precision in Scientific Notation

Take a look at these two measurements. Do you know the difference between the two numbers?

1.5 X 102 meters

1.50 X 102 meters

Are they equivalent? Perhaps they are, perhaps not. The first number is written a little less precisely than the second. The first number, 1.5 X 102, has 2 significant figures. The number can also be written in traditional notation as 150, with no trailing decimal point. This implies that it is accurate to the nearest 10 meters. The second number, 1.50 X 102, has three significant figures. The trailing zero implies that is is accurate to the nearest meter. This number can also be written as 150., with a trailing decimal point. In science and engineering, these two similar-looking numbers are indeed very different.

Therefore, when we write 150 meters (with no trailing decimal point) we are implying that the measurement can be anywhere from 145 to 154 meters. When we write 150. meters (with a trailing decimal point) we are implying that our measurement is greater than or equal to 149.5 meters and less than 150.499 (repeating) meters.

What if we write "150.0 meters"? Well, in this case we are implying that our measurement is greater than or equal to 150.0 meters and less than 150.05 meters.


Scientific Calculator
Scientific Calculator

Basic Arthmetic in Scientific Notation

We can add numbers that are written in scientific notation:

    1.0 X 1012
 +  2.0 X 1012
 _____________
    3.0 X 1012


>>

It's easy: line up the numbers on the decimal point and make sure the exponents are identical. Then, add the numbers to the left of the multiplication symbol and simply bring down the exponent. If the exponents don't match, you can reformat one or both numbers in order to get them to match. For example:

    1.0 X 1011
 +  2.0 X 1012
 _____________
 


>>

We can't perform this addition without massaging the numbers a little. The decimal points appear to line up, but exponents don't match. To fix this problem all we have to do is adjust the top number by incrementing the exponent and at the same time decreasing the number by an order of magnitude:

    0.1 X 1012
 +  2.0 X 1012
 _____________
    2.1 X 1012


>>

Very easy! The trick is to notice that

0.1 X 1012

is exactly the same as

1.0 X 1011



Instead, we can use scientific notation to write the number as:

1 X 1012

which is much easier to read and write. There are many ways to write the same number in scientific notation. We could also write the above number as:

10 X 1011

which is an equivalent value, but a slightly different representation. It's acceptable, but it's not normalized. A special form of scientific notation, referred to as normalized scientific notation, requires that the absolute value to the left of the multiplication symbol must be greater than or equal to 1 and less than 10. The first example, above, is in normalized scientific notation form.

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Comments 14 comments

The Old Firm profile image

The Old Firm 7 years ago from Waikato/Bay Of Plenty, New Zealand

I'm afraid that I must disagree with you, at least in part. I'm doing so not on any deep and extensive mathematical background, but on what I perceive is practical reality when I read a series of numbers. We may then well be approaching the discussion using different parameters, and therefore both right. Again my interpretation may be open to refutal (If you wish ridicule)

"Therefore, when we write 150 meters (with no trailing decimal point) we are implying that the measurement can be anywhere from 150 to 159 meters." - If I read that figure, with or without the trailing decimal, the zero would imply to me that the actual distance was closer than half a unit to exactly 150. That is >149.5 & <150.5. greater than this it would be shown as 151, or whichever whole number it most closely approached. If a decimal was introduced the same argument would apply to the final unit following the decimal.

The problem appears to me to be that in the practical world a decimal followed with zip in the context you described here has no relevance. Certainly, once that you introduce exponents into the equation it's a whole new ball game.

Just to show we do read these hubs,

Cheers, TOF

PS, wanna buy a watch?


nicomp profile image

nicomp 7 years ago from Ohio, USA Author

Are you stalking me? Man, I hope so. We engineers got to stick together.

I think if we write 150 (no trailing decimal), we are implying that the zero is not significant. if we write 151(still no trailing decimal), we are implying that the rightmost one is significant. We have two examples with no decimal point but with different interpretations of the digit in the ones position.

If the digit in the ones position is not significant, then the measurement is to the nearest 10 meters, so I guess it's actually in the range 145 meters to 154 meters.

The key difference seems to be the different interpretations of the rightmost zero in the absence of a decimal point. Many sources agree that this interpretation varies in the "real world", so I think I agree with your thesis.

I looked on a few other sites, including the Repository of all Knowledge, Wikipedia, and they wavered on the subject as well.

Do you happen to have the correct time? This dang watch stopped again...


The Old Firm profile image

The Old Firm 7 years ago from Waikato/Bay Of Plenty, New Zealand

Just a tick, I'll check my cell-phone........bugger the batteries flat.

There's no moon, hang on, the cat wants to go out, it must be 1330zulu


The Old Firm profile image

The Old Firm 7 years ago from Waikato/Bay Of Plenty, New Zealand

Replying to your query on my home page, I can do a hub on most things, whether it makes any sense or not is a different matter.

If by static you imply balanced, that's pretty much physics 101 isn't it? The sort of stuff I did as a teen when learning the principles of aerodynamics - lift resolved as vertical lift and induced drag, and similar arrow in a rectangle basics if I can remember back to 1962 correctly.

Mind you, that comment should throw anyone reading my fan club into a first class tail-spin. Cheers mate. TOF


Lgali profile image

Lgali 7 years ago

very unique hub thanks


NATE 7 years ago

Not what i wanted...


nicomp profile image

nicomp 7 years ago from Ohio, USA Author

Sorry, Nate! Let me know what yo do want and I'll work on it for you.


The girl 6 years ago

im only in 5th grade and you make this so much harder than it should be so make it were we can under stand it.


nicomp profile image

nicomp 6 years ago from Ohio, USA Author

@The girl says: Sorry! if you don't understand it, how do you know it could be made easier?


nicomp profile image

nicomp 5 years ago from Ohio, USA Author

Well, it's been 11 months, so "the girl" must be in 6th grade by now. Wonder if she understands this hub yet?


saurav thapa 5 years ago

how can you such great engineers get stuck in small small things... you guys are fighting for what is worthless.


anemometers profile image

anemometers 5 years ago from Anemometer Installation Areas

great hub. well done


???? 5 years ago

I WAS NOT HELPED AT ALL BY THIS C.R.A.P!!


nicomp profile image

nicomp 5 years ago from Ohio, USA Author

@????: that's an acronym?

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