Standard Form to Scientific Form
Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik’s cube) and very small numbers such as 0.000000000000000000000340 (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10.
To see how this is done, let us start with a somewhat smaller number such as 2480. See Figure 1.
The last form in Figure 1 is called the scientific form of the number. There is one nonzero digit to the left of the decimal point and the absolute value of the exponent on 10 records the number of places the original decimal point was moved to the left. If instead we have a very small number, such as 0.00059, we instead move the decimal place to the right, as in the following:
There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right.
Writing a Number in Scientific Notation
To write a number in scientific notation:
- Move the decimal point so that there is one nonzero digit to its left.
- Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.
A number written in scientific notation can be converted to standard form by reversing the process described above.
Normalized Scientific Notation
Any given number can be written in the form of a×10b in many ways; for example, 350 can be written as 3.5×102 or 35×101 or 350×100. In normalized scientific notation, the exponent b is chosen so that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). Following these rules, 350 would always be written as 3.5×102. This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation, the exponent b is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as −5×10−1). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 cannot be written in normalized scientific notation since it cannot be expressed as a×10b for any non-zero a. Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as engineering notation, is desired. Normalized scientific notation is often called exponential notation—although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (as in 315× 220).
Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 107 cannot always be conveniently displayed, the letter E or e is often used to represent "times ten raised to the power of" (which would be written as "x 10b") and is followed by the value of the exponent. Note that in this usage the character e is not related to the mathematical constant e or the exponential function ex (a confusion that is less likely with capital E), and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation. The use of this notation is not encouraged by publications.