# 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 10** ^{2 }**meters

1.50 X 10** ^{2}** 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
10** ^{2}**, 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 10

**, 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.**

^{2}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.

## Basic Arthmetic in Scientific Notation

We can add numbers that are written in scientific notation:

1.0 X 10^{12}

+ 2.0 X 10^{12}

_____________

3.0 X 10^{12}

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 10^{11}

+ 2.0 X 10^{12}

_____________

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 10^{12}

+ 2.0 X 10^{12}

_____________

2.1 X 10^{12}

Very easy! The trick is to notice that

0.1 X 10^{12 }

is exactly the same as

1.0 X 10^{11}

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

1 X 10^{12}

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 10^{11}

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

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?

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

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

very unique hub thanks

Not what i wanted...

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.

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

great hub. well done

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

14