Scientific Notation

Scientific notation is a method of writing very large and very small numbers. Ordinary numbers are useful for everyday measurement, such as daily temperatures and automobile speeds, but for large measurements like astronomical distances, scientific notation provides a way to express these numbers in a short and concise way.

The basis of scientific notation is the power of ten. Since many large and small numbers have a few integers with many zeros, the power of ten can be used to shorten the length of the written number.

A number written in scientific notation has two parts. The first part is a number between 1 and 10, and the second part is a power of ten. Mathematically, writing a number in scientific notation is the expression of the number in the form n × 10^x where n is a number greater than 1 but less than 10 and x is an exponent of 10. An example is 15,653 written as 1.5653 × 10⁴. This type of notation is also used for very small numbers such as 0.0000000072, which, in scientific notation, is rewritten as 7.2 × 10⁻⁹.

Some examples of measurements where scientific notation becomes useful follow.

• The wavelength for violet light is 40-millionths of a centimeter 4 × 10⁻⁵cm.
• Some black holes are measured by the amount of solar masses they could contain. One black hole was measured as 10,000,000 or 1.0 × 10⁷ solar masses .
• The weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms (6.645 × 10⁻²⁷ kilograms).
• A computer hard disk could hold 4 gigabytes (about 4,000,000,000 bytes) of information. That is 4.0 × 10⁹ bytes.
• Computer calculation speeds are often measured in nanoseconds. A nanosecond is 0.000000001 seconds, or 1.0 × 10⁻⁹ seconds.

Converting Numbers into Scientific Notation

Complete the following steps to convert large and small numbers into scientific notation. First, identify the significant digits and move the decimal place to the right or left so that only one integer is on the left side of the decimal. Rewrite the number with the new decimal place and include only the identified significant digits. Then, following the number, write a multiplication sign and the number 10. Raise the 10 to the exponent that represents the number of places you moved the decimal point.

If the number is large and you moved the decimal point to the left, the exponent is positive. Conversely, if the number is small and you moved the decimal point to the right, the exponent is negative.

If a chemist is trying to discuss the number of electrons expected in a sample of atoms, the number may be 1,100,000,000,000,000,000,000,000,000. Using three significant figures the number is written 1.10 × 10²⁷. Along the same lines the weight of a particular chemical may be 0.0000000000000000000721 grams. In scientific notation the example would be written 7.21 × 10⁻²⁰

see also Measurement, Metric System of; Numbers, Massive; Powers and Exponents.

Brook E. Hall

Bibliography

Devlin, Keith J. Mathematics, the Science of Patterns: The Search for Order in Life, Mind, and the Universe. New York: Scientific American Library, 1997.

Morrison, Philip, and Phylis Morrison. Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding Another Zero. New York: Scientific American Library, 1994.

Pickover, Clifford A. Wonders of Numbers: Adventures in Math, Mind, and Meaning. Oxford, U.K.: Oxford University Press, 2001.

POWERS, EXPONENTS, AND SCIENTIFIC NOTATION

The process of calculating scientific notation comes from the rules of exponents.

• Any number raised to the power of zero is one.
• Any number raised to the power of one is equal to itself.
• Any number raised to the n th power is equal to the product of the number used as a factor n times.

scientific notation means of expressing very large or very small numbers in a compact form that is easy to use in computations. In this notation, any number is expressed as a number between 1 and 10 multiplied by a power of 10 that indicates the correct position of the decimal point in the original number; numbers greater than 10 are expressed by positive powers of 10 and numbers less than 1 are expressed by negative powers of 10 (see exponent ). For example, 43,700 is written in scientific notation as 4.37 × 10 ⁴ and 0.00526 as 5.26 × 10 ^-3 . The larger the converted number, the more compactness is achieved: for example, the speed of light, about 30,000,000,000 cm per sec, becomes 3 × 10 ¹⁰ cm per sec. Calculations are greatly simplified by use of scientific notation: the first parts of a pair of numbers to be multiplied or divided are combined manually or by slide rule and the powers of 10 are added or subtracted in accordance with the rules for exponents. If the first part of the result is greater than 10, an adjustment is made. For example, in order to multiply 832,000 by 0.00035, one converts first to scientific notation as follows: (832,000)×(0.00035)=(8.32×10 ⁵ )×(3.5×10 ^-4 )=8.32×3.5×10 ⁵ ×10 ^-4 =29.12×10 ¹ =2.912×10 ² (in scientific notation) or 291.2 (in ordinary notation).