# scientific notation (power-of-10 notation)

Scientific notation, also called power-of-10 notation, is a method of writing extremely large and small numbers. There are two forms of this scheme; one is by far more common.

In common scientific notation, any nonzero quantity can be expressed in two parts: a coefficient whose absolute value is greater than or equal to 1 but less than 10, and a power of 10 by which the coefficient is multiplied. In some writings, the coefficients are closer to zero by one order of magnitude. In this scheme, any nonzero quantity is expressed in two parts: a coefficient whose absolute value is greater than or equal to 0.1 but less than 1, and a power of 10 by which the coefficient is multiplied. The quantity zero is denoted as 0 unless precision is demanded, in which case the requisite number of significant digits are written out -- for example, 0.00000.

For numbers of reasonable magnitude, conventional decimal notation is often used, even in scientific writings. Let *s* be a number rounded or truncated to a few significant figures. If the absolute value of *s* is at least 0.001 (10 ^{-3} ) but less than 10,000 (10 ^{4} ), then *s* is usually written out in full. Examples are 21.3389 and -0.002355. However, if the absolute value of *s* is smaller than 0.001 or if it is 10,000 or larger, scientific notation is usually preferred, because writing such numbers out in decimal form can be confusing and messy. This is especially true when the absolute value of *s* is very close to zero or is exceedingly large. It is inconvenient, for example, to write out either of the expressions 6.0205 x 10 ^{74} or -0.64453 x 10 ^{-45} in decimal form.

The table shows several examples of numbers written in standard decimal notation (left-hand column) and in scientific notation (right-hand column). For negative numbers, the values are simply mirror-image positive numbers; a minus sign is placed in front of the values. The number of digits in the coefficient is the number of significant figures. Note that an expression can have various degrees of precision; the greater the number of significant figures, the greater the precision.

Number in decimal form |
Examples in scientific notation |

1,222,000.00 | 1.222 x 10 ^{6} 1.22200000 x 10 ^{6} 0.1222 x 10 ^{7} 0.122200000 x 10 ^{7} |

0.00003450000 | 3.45 x 10 ^{-5} 3.450000 x 10 ^{-5} 0.345 x 10 ^{-4} 0.3450000 x 10 ^{-4} |

-9,876,543,210 | -9.87654 x 10 ^{9} (approximately)-9.876543210 x 10 ^{9} (exactly)-0.987654 x 10 ^{10} (approximately)-0.9876543210 x 10 ^{10} (exactly) |

-0.0000000100 | -10 ^{-8} -1.00 x 10 ^{-8} -0.1 x 10 ^{-7} -0.100 x 10 ^{-7} |

Scientific notation makes it easy to multiply and divide gigantic and/or minuscule numbers, when the use of decimal notation would give rise to frustration. Consider, for example, the following product:

2.56 x 10 ^{67} x -8.33 x 10 ^{-54}

To obtain the product of these two numbers, the coefficients are multiplied, and the powers of 10 are added. This produces the following result:

2.56 x (-8.33) x 10 ^{67+(-54)}

= 2.56 x (-8.33) x 10 ^{67-54}

= -21.3248 x 10 ^{13}

The proper form of common scientific notation requires that the absolute value of the coefficient be larger than 1 and less than 10. Thus, the coefficient in the above expression should be divided by 10 and the power of 10 increased by one, giving:

-2.13248 x 10 ^{14}

Because both multiplicands in the original product are specified to only three significant figures, a scientist might see fit to round off the final expression to three significant figures as well, yielding:

-2.13 x 10 ^{14}

as the product.

Now consider the quotient of the two numbers multiplied in the previous example:

(2.56 x 10 ^{67} ) / (-8.33 x 10 ^{-54} )

To obtain the quotient, the coefficients are divided, and the powers of 10 are subtracted. This gives the following:

(2.56 / (-8.33)) x 10 ^{67-(-54)}

= (2.56 / (-8.33)) x 10 ^{67+54}

= -0.30732 x 10 ^{121}

The proper form of common scientific notation requires that the absolute value of the coefficient be larger than 1 and less than 10. Thus, the coefficient in the above expression should be multiplied by 10 and the power of 10 decreased by one, giving:

-3.0732 x 10 ^{120}

Because both numbers in the original quotient are specified to only three significant figures, a scientist might see fit to round off the final expression to three significant figures as well, yielding:

-3.07 x 10 ^{120}

as the quotient.

Also see order of magnitude and significant figures .

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