Exponential notation is a convenient way to express large and small numbers. Numbers can be expressed as a coefficient multiplied by a base raised to an exponent (power).

A ื B^{N}

A = coefficient

B = base

N = exponent

The coefficient is not always written explicitly if it happens to be equal to one. The exponent represents the number of times the base number is multiplied by itself. If the exponent is negative the expression represents a reciprocal. The examples below all have A = 1.

2^{3} = 2ื2ื2 = 8

2^{3} = 1/2^{3} = 1/8 = 0.125

5^{4} = 5ื5ื5ื5 = 625

5^{4} = 1/5^{4} = 1/625 = 0.0016

10^{6} = 10ื10ื10ื10ื10ื10 = 1000000

10^{6} = 1/10^{6} = 1/1000000 = 0.000001

Expressing numbers in the binary system with base-2 is sometimes convenient for work in computer science. Other bases may be useful in particular specialized fields as well. In daily life the decimal system with base-10 is used most often. Thus using exponential notation with base-10 would offer the most advantages and convenience in many applications. Numbers expressed in exponential notation with base-10 using integer exponents are said to be written in scientific notation. In general, exponents need not be integers. Recall that roots are equivalent to fractional exponents.

In chemistry, as in daily life, the decimal system is used most commonly. More importantly, extremely large and small numbers are frequently encountered in chemistry. Scientific notation is commonly used by chemists for these reasons. Any number may be expressed in scientific notation using the following form:

A ื 10^{N}

A = coefficient, any number from 1-10

N = exponent, any integer (positive, negative, or zero)

Below are several examples that demonstrate the equivalence of scientific notation and standard decimal notation. As the examples show, the larger the magnitude of the exponent the more convenient it is to use scientific notation and avoid writing many zeros.

Standard decimal notation | Scientific notation |
---|---|

300,000,000 | 3ื10^{8} |

6,000,000,000,000 | 6ื10^{12} |

6,700,000,000,000 | 6.7ื10^{12} |

602,000,000,000,000,000,000,000 | 6.02ื10^{23} |

0.000000000000000000009 | 9ื10^{21} |

0.00000005 | 5ื10^{8} |

0.0000000500 | 5.00ื10^{8} |

How were these equivalent expressions determined? What is needed is an algorithm, or procedure, for doing the conversion of numbers to and from scientific notation.

1) Move the decimal point so that it follows the first non-zero digit to generate a number from 1-10. Count the number of places moved to the left or right.

2) Write the number generated in step one multiplied by 10B where B is the number of places the decimal point was moved.

3) Make sure the sign of the exponent is correct. If the decimal was moved left, the sign is positive. If the decimal was moved right, the sign is negative.

Below are two examples. In each example, the degree symbol บ is used to represent the final position of the decimal point. The original position of the decimal is indicated to show how many places the decimal was moved. Spaces are used in place of commas in large numbers. This notation will be used throughout this section.

A) Convert 140,000 to scientific notation.

Step 1) 1บ40 000. (moved 5 places to the left)

Step 2) 1.4ื10^{5}

Step 3) Exponent is positive because the decimal was moved left.
The answer is 1.4ื10^{5}

B) Convert 0.00082 to scientific notation.

Step 1) 0.0008บ2 (moved 4 places to the right)

Step 2) 8.2ื10^{4}

Step 3) Exponent is negative because the decimal was moved right.

The answer is 8.2ื10^{4}

Procedure to convert from scientific notation to a standard decimal number:

1) Move the decimal point as many places as indicated by the exponent. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. Add zeros to act as place holders if necessary after moving the decimal point.

2) Write the number generated in step one without the power of ten.

A) Convert 2.85ื10^{4} to standard decimal notation.

Step 1) 2.8500บ ื10^{4} (moved 4 places to the right)

Step 2) 28 500

The answer is 28,500

B) Convert 1.61ื10^{19} to standard decimal notation.

Step 1) 0บ0000000000000000001.61ื10^{19} (moved 19 places to the left)

Step 2) 0.000000000000000000161

The answer is 0.000000000000000000161

Any number can be expressed in scientific notation using the appropriate exponent, or power of 10. When using extremely large and extremely small numbers, it is often convenient to limit the exponent to a multiple of three. When this is done the notation is called engineering notation. The result is that when written in engineering notation, the number of thousands, millions, trillions, millionths, billionths, etc. is readily apparent.

A ื 10^{N} A = coefficient, any number from 1-1000

N = exponent, integer that is a multiple of three

Examples: 5 million = 5,000,000 = 5ื1,000,000 = 5ื10^{6}

15.6 trillion = 15,600,000,000,000 = 15.6ื1,000,000,000,000 = 15.6ื10^{12}

2 thousandths = 0.002 = 2ื0.001 = 2ื10^{3}

350 millionths = 0.000350 = 350ื10^{6}

Note that 15.6ื10^{12} (engineering notation) is the same as 1.56ื10^{13} (scientific notation).

The procedures for converting a number to/from engineering notation are essentially the same as those for converting to/from scientific notation. The difference is that A can be a number from 1-1000 and the exponent B will always be a multiple of three (i.e. the decimal point is always moved three places at a time).

Most chemistry textbooks use scientific notation rather than engineering notation, but engineering notation does have two advantages. When using metric units, the only prefixes used when the magnitude of the exponent is three or larger are multiples of three (e.g. kilo, mega, giga, micro, nano, etc.). Another advantage of engineering notation is that it can make certain calculations easier when done by hand. When using a calculator, the calculator can generally be set to a mode that gives answers in either scientific or engineering notation so no additional work by students is necessary.

In general, a number can be written in exponential notation in any number of equivalent ways with any number of different coefficients. When the coefficient, A, is from 1-1000 it is called engineering notation. When the coefficient is from 1-10 it is called scientific notation. The coefficient can also be from 0-1. This would be acceptable in generic exponential notation. In solving particular problems it is often convenient to be able to change the exponent.

Procedure to convert from exponential notation to scientific notation: 1) Move the decimal point to generate a number from 1-10. Count the number of places moved to the left or right. 2) Adjust the exponent. If the decimal was moved left, increase the exponent by the number of places. If the decimal was moved right, decrease the exponent.

A) Convert 36.9ื10^{6} to scientific notation.

Step 1) 3บ6.9ื10^{6} (moved 1 place to the left)

Step 2) 3.69ื10^{7}

The answer is 3.69ื10^{7}

B) Convert 0.0014ื10^{3} to scientific notation.

Step 1) 0.001บ4ื10^{3} (moved 3 places to the right)

Step 2) 1.4ื10^{6}

The answer is 1.4ื10^{6}

Any mathematical operation that can be performed on a number expressed in standard decimal notation can be performed on a number written in any form of exponential notation, including scientific and engineering notation. It is often true that these operations are more easily performed in scientific notation, especially when doing calculations by hand. Multiplication, division, and logarithms are generally more convenient using scientific notation. Addition and subtraction are often less convenient.

The concept order of magnitude refers to the exponent from a number written in exponential notation. One million is 106, and any number of the form Aื106 is in the millions. The order of magnitude is equal to six (though a salary in the millions is called a seven figure salary). If the value of A is an integer number of millions it will have six zeros, though, e.g. 2ื10^{6}, 3ื10^{6} and 4ื10^{6} each have six zeros). If A is a decimal, there will be fewer than six zeros, e.g. 2.5ื10^{6} and 9.8ื10^{6} each have five zeros. Calculating a logarithm provides information about order of magnitude.

Multiplication and division operations performed on numbers expressed in scientific notation are often simpler to perform than on standard numbers. When very large and small numbers are being used, scientific notation allows one to multiply or divide small numbers less than ten and then add or subtract the exponents.

Procedure to multiply numbers expressed in scientific notation

1) Regroup so that the coefficients are together and the exponents are together

2) Multiply the coefficients. Add the exponents to generate the new exponent

3) If needed, change the exponent so that the coefficient is from 1-10 (Section 4.2a)

A) Calculate z = (3.0ื10^{6}) ื (2.5ื10^{3})

Step 1) z = (3.0ื2.5) ื (10^{6}ื10^{3})

Step 2) z = 7.5ื10^{(6+3)}

The answer is 7.5ื10^{9}

B) Calculate z = (4.0ื10^{3}) ื (3.0ื10^{4})

Step 1) z = (4.0ื3.0) ื (10^{3}ื10^{4})

Step 2) z = 12ื10^{(3+4)} = 12ื10^{7}

Step 3) z = 12ื10^{7} = 1บ2.ื10^{7} = 1.2ื10^{8}

The answer is 1.2ื10^{8}

Procedure to divide numbers expressed in scientific notation

1) Regroup so that the coefficients are together and the exponents are together

2) Divide the coefficients. Subract the exponents to generate the new exponent

3) If needed, change the exponent so that the coefficient is from 1-10 (Section 4.2a)

A) Calculate z = (8.0ื10^{6}) / (2.0ื10^{4})

Step 1) z = (8.0/2.0) ื (10^{6}/10^{4})

Step 2) z = 4.0ื10^{(64)}

The answer is 4.0ื10^{2}

B) Calculate z = (2.8ื10^{8}) / (5.0ื10^{3})

Step 1) z = (2.8/5.0) ื (10^{8}ื10^{3})

Step 2) z = 0.56ื10^{(83)} = 0.56ื10^{5}

Step 3) z = 0.56ื10^{5} = 0.5บ6ื10^{5} = 5.6ื10^{4}

The answer is 5.6ื10^{4}

Addition and subtraction operations performed on numbers expressed in scientific notation are often less simple to perform than on standard numbers. Both numbers need to have a common exponent. Usually the number with the smaller exponent is converted to have the larger exponent. After exponents have been changed, the addition or subtraction can proceed normally.

Procedure to add or subtract numbers expressed in scientific notation

1) If needed, change one exponent so that both are equal

2) Align the decimals and add or subtract

3) If needed, change the exponent so that the coefficient is from 1-10

A) Calculate (8.00ื10^{6}) + (2.0ื10^{4})

Step 1) 2.0ื10^{4} = 0บ02.0ื10^{4} = 0.020ื10^{6}

Step 2) 8.00 ื10^{6}

+ 0.020 ื10^{6}

8.02ื10^{6}

The answer is 8.02ื10^{6}

B) Calculate (2.8ื10^{4}) (5ื10^{3})

Step 1) 5ื10^{3} = 0บ5.ื10^{3} = 0.5ื10^{4}

Step 2) 2.8ื10^{4}

0.5ื10^{4}

2.3ื10^{4}

The answer is 2.3ื10^{4}

Metric and SI units are based on decimal conversions. Metric prefixes represent specific powers of 10. Thus, measurements written in scientific notation or engineering notation can easily be converted from base units to units with a prefix and vice versa. All the prefixes corresponding to exponents equal to three or larger have equivalents that are multiples of three.

Below is a table of some of the common metric prefixes with the numerical equivalents.

Prefix | Numerical equivalent | Equivalent power of 10 |
---|---|---|

Giga (G) | 1,000,000,000 | 10^{9} |

Mega (M) | 1,000,000 | 10^{6} |

Kilo (k) | 1,000 | 10^{3} |

Hecto (h) | 100 | 10^{2} |

Deka (da) | 10 | 10^{1} |

- | 1 | 10^{0} |

Deci (d) | 0.1 | 10^{-1} |

Centi (c) | 0.01 | 10^{-2} |

Milli (m) | 0.001 | 10^{-3} |

Micro (ต) | 0.000001 | 10^{-6} |

Nano (n) | 0.000000001 | 10^{-9} |

Scientific and engineering notation can be used to give a convenient equivalent expression for metric units that include a prefix. The prefix and ื10N can directly substitute for each other. Below are some examples.

1.5 kg = 1.5ื10^{3} g

2.54ื10^{2} m = 2.54 cm

454 nm = 454ื10^{9} m

8.4ื10^{3} L = 8.4 mL