Scientific Notation

There is some ambiguity in nice round numbers like 7000, because you don't know how many significant digits it represents. Is the 7000 a rounding up from 6999, 6991, or 6911 -- or is it EXACTLY 7000?

Scientists would take care of that ambiguity by writing

7.00 x 103, 7.0 x 103, 7 x 103 -- or 7.000 x 103. The last one has four significant figures.

Engineers write it as 7E3. (Note that the "3" is NOT a superscript in engineering notation, as it is with scientific notation.)

Be careful here, nothing is trivial or obvious. On one level this question seems trivial, but on other level it is not. Why? In the domain of representing numbers in pure mathematics, they can be considered as being represented Exactly by either an unlimited number of digits via say a decimal expansion, think of the the rational number 1/3. The exact representation is 1/3, but what about the decimal expansion?

Once we come into the REAL world where practical measurement come in the number representations are not so clear cut and the concept of significant digits comes into its own, that is digits we are confident in our measuring. I mean is the 7000, 7000.000 if it is then in scientific notation its 7.000 X 10 to the power of 3 not 7 x 10 to the power of 3. I hope this makes sense many of my university students have trouble with significant figures, this is what this example was trying to get across, a deeper idea of the conflict between the ideal world and the practical measuring world.

It is 1.000*10^7.

The number could be written as 1*10^7: the trailing zeros are to show the level of accuracy (or degree of rounding).

Scientific notation is normally used for numbers that are either far to large or far to small to be written conveniently in decimal notation.A,B

For example the Earth's mass is approximately: 5,973,600,000,000,000,000,000,000.0 kg

In scientific notation this would be written as:

5.9736 x 1024 kg.

In normalised scientific notation numbers are written in the form:A,B

a x 10n

Where:

a is a number between 1 and 10

n is a positive or negative whole number.

In engineering notation, the n value is commonly in the form of multiples of 3. In this way the number will always explicitly match the corresponding SI prefixes.B

For example a distance of 50,000 m would be written as:

Scientific Notation: 5 x 104 m

Engineering notation: 50 x 103 m

In this example 103 corresponds to the SI prefix "kilo"C as such the engineering notation could be directly described verbally as "fifty kilometres" whereas scientific notation yields the much more unwieldy "five times ten to the power four metres" which is much less intuitively easy to understand, even though it is exactly the same distance.

Guidance on converting to and from scientific notation is given in the related links. Specifically References A and B.

References:

A Scientific notation - Engineering Maths Help from the 'mathcentre' Academic Website.

B Scientific notation: Wikipedia Entry.

C List of SI prefixes: Wikipedia Entry.

Please see related links.

Scientific notation is a way to express numbers that are either very small or very large. In traditional notation the first kind would have a lot of 0s between the decimal point and the first significant figure whereas the second kind would have a large number of trailing 0s. The need for scientific notation arose from advances in various branches of science: atomic particles in physics or chemistry, astronomical or cosmological distances, size of single cell animals. Nowadays, even non-scientific values such as population, national debts (of some countries) could usefully utilize scientific notation.

Scientific notation is a way of representing numbers, in the form

a*10b where 1 ≤ |a| < 10 is a decimal number and b is an integer (negative or positive).

a is called the mantissa and b is called the exponent.