Engineering

Scientific notation is a way to "easily" or "conveniently" write very large or very small numbers. As these numbers are frequently encountered in the sciences, the term scientific notation was introduced to name this "neat" way to "package" these quantities so that they might be more easily grasped and understood.

Scientific notation is a useful way of dealing with very large and very small numbers. It allows them to be presented in a form where their magnitude can be seen more easily. Also it can simplify calculations by allowing you to concentrate on the significant digits rather than the orders of magnitude which are very easily dealt with. This latter advantage has somewhat diminished with the widespread availability of calculators and computers. But previously, people used log tables and slide rules for multiplication and division. These calculating devices depended on thinking of numbers in their scientific notation and utilizing the significant digits.

The idea behind scientific notation is to write numbers in terms of powers of ten - either positive (for very large numbers), or negative (for very small ones). As an example, consider the mass of an electron, which is approximately 0.0000000000000000000000000001 grams. An easier way to write it uses the significant digit 1 and an exponent based on a multiple of ten. The number becomes the easily represented 1 x 10-28 g.

The simple rule is to take your "numbers" and move the decimal point to the left or right so that only one figure is to the left of the decimal. Then write the rest of the significant digits to the right of the decimal, and tack on the appropriate power of ten (again, either positive or negative) to restore the proper value to the figure.

Scientific Notation also avoids the headache and potential errors of counting lots of zeros.

The number 123000000 in scientific notation is written as:

1.23 x 108

The first number 1.23 is called the coefficient. It is always a single digit followed by a decimal point and then the rest, but usually only two digits.

The second number is called the base and in scientific notation must always be 10. In the number 1.23 x 108 the number 8 is the exponent or power of ten.

For large numbers :

1) Put the decimal after the first digit and drop the zeroes. In the number 123,000,000 the coefficient will be 1.23

2) Then write the times "x" and the base 10.

3) To find the exponent count the number of places from the "new" decimal point to the end of the number. In 123000000 there are 8 places. Therefore the exponent is 8.

There are some minor variations that have evolved to fill different needs, usually because not all fonts or printers allow superscripts: 123000000 can be written as:

1.23 E+11 or 1.23 X 10^11 or 1.23 x 1011

For small numbers :

For numbers less than one we use a similar approach. These numbers all have negative exponents. For example 0.00000123 second (1.23 microseconds) is written:

1.23 E-6 or 1.23 x 10^-6 or 1.23 x 10-6

Take the original number 0.00000123 and shift the decimal point to the right until you get the coefficient in proper form, as above. The number of digits shifted is then the negative exponent.

Notes:

a) Numbers less than one all use negative exponents, but what about negative numbers, such as -0.04? We can write this as

-4.0 x 10-2

b) Always make sure the E is capitalized in 1.23 E-6, otherwise it can be confused with "e" the base of the natural log system.

c) Some scientific and engineering fields have special rules, such as electronics where scientific notation is usually in powers divisible by three, such as -3, 3, 6, 9, 12, etc. This is because electronic components are made using standard SI prefixes such as kilo, micro, nano, or pico.

d) Usually, Scientific Notation is ignored if you want to keep numbers in common formats, such as 315 microseconds, instead of 3.15 x 10-4 seconds, but this is a matter of preference.

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.