SCIENTIFIC NOTATION

Because of the very large and very small numbers frequently encountered in most scientific disciplines it is both convenient and economical in terms of time and space to express numbers in so-called scientific notation rather than conventional notation. For example, the mass of the sun can either be expressed as 1.989×10³³ grams in scientific notation or 1,989,000,000,000,000,000,000,000,000,000,000 grams in conventional notation. The mass of an electron is expressed as
9.110×10^-28 grams in scientific notation or 0.000 000 000 000 000 000 000 000 000 9110 grams in conventional notation.

There is a simple rule for remembering what the exponent (the small superscripted number, including its sign) of ten means in scientific notation; it is simply the number of places one would shift the decimal point, positive to the right or negative to the left, to convert the number from scientific to conventional notation, e.g.,

4.33×10⁴ means 43300.= 43300, the decimal is shifted four places to the right in converting the number to conventional notation, or

4.33×10^-3 means 0.00433, the decimal is shifted three places to the left in converting the number to conventional notation, or

4.33×10⁰ means 4.33, the decimal is not shifted at all.

To multiply numbers written in scientific notation simply multiply in the usual way, but add the attached exponents, e.g.,

(4×10⁷)×(2×10⁵) = 4×2×10⁷⁺⁵ = 8×10¹², or

(6×10^-3)×(4×10⁸) = 6×4×10^-3+8 = 24×10⁵ = 2.4×10⁶.

To divide numbers written in scientific notation simply divide in the usual way, but subtract the exponent of the divisor from that of the dividend, e.g.,

(4×10⁷)/(2×10⁵) = (4/2)×10^7-5 = 2×10², or

(6×10^-3)/(4×10⁸) = (6/4)×10^-3-8 = 1.5×10^-11, or

(4×10⁸)/(6×10^-3) = (4/6)×10^8-(-3) = 0.667×10¹¹ = 6.67×10¹⁰.

To add or subtract a pair of numbers written in scientific notation it is necessary that both numbers be represented in terms of a common power of ten just as a pair of fractions which are to be added or subtracted must have a common denominator. Often this requires that one of the numbers be converted to a form represented by a different power of ten.  For example, to accomplish the addition 2.61×10⁵ + 1.47×10⁴, we note that 1.47×10⁴ = 0.147×10⁵, so 2.61×10⁵ + 1.47×10⁴ = 2.61×10⁵ + 0.147×10⁵ = (2.61 + 0.147)×10⁵  = 2.76×10⁵.  Alternatively, we note that 2.61×10⁵ = 26.1×10⁴, so 2.61×10⁵ + 1.47×10⁴ = 26.1×10⁴ + 1.47×10⁴ = 
(26.1 + 1.47)×10⁴ = 27.6×10⁴ = 2.76×10⁵.