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Power of Ten Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Standard and Alternative Form

Scientists and engineers like to express extreme numerical values using an exponential technique known as power-of-10 notation. This is usually what is meant when they talk about scientific notation.

Standard Form

A numeral in standard power-of-10 notation is written as follows:

m.n × 10 z

where the dot (.) is a period, written on the base line (not a raised dot indicating multiplication), and is called the radix point or decimal point. The value m (to the left of the radix point) is a positive integer from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. The value n (to the right of the radix point) is a non-negative integer. The value z, which is the power of 10, can be any integer: positive, negative, or zero. Here are some examples of numbers written in standard scientific notation:

Scientific Notation Power-of-10 Notation Standard Form

Alternative Form

In certain countries, and in many books and papers written before the middle of the 20th century, a slight variation on the above theme is used. The alternative power-of-10 notation requires that m be 0 rather than 1, 2, 3, 4, 5, 6, 7, 8, or 9. When the above quantities are expressed this way, they appear as decimal fractions larger than 0 but less than 1, and the value of the exponent is increased by 1 compared with the standard form:

Scientific Notation Power-of-10 Notation Alternative Form

These are the same three numerical values as the previous three; the only difference is the way they’re expressed. It’s like saying you’re driving down a road at 50,000 meters per hour rather than at 50 kilometers per hour.

The “Times Sign”

The multiplication sign in a power-of-10 expression can be denoted in various ways. Most scientists in America use the cross symbol (x), as in the examples shown above. But a small dot raised above the base line (·) is sometimes used to represent multiplication in power-of-10 notation. When written that way, the above numbers look like this in the standard form:

Scientific Notation Power-of-10 Notation The “times Sign”

This small dot should not be confused with a radix point, as in the expression

m.n · 10 z

in which the dot between m and n is a radix point and lies along the base line, while the dot between n and 10 z is a multiplication symbol and lies above the base line. The small dot is preferred when multiplication is required to express the dimensions of a physical unit. An example is the kilogrammeter per second squared, which is symbolized kg · m/s 2 or kg · m · s –2 .

When using an old-fashioned typewriter, or in word processors that lack a good repertoire of symbols, the lowercase, non-italicized letter x can be used to indicate multiplication. But this can cause confusion, because it’s easy to mistake this letter x for a variable. So in general, it’s a bad idea to use the letter x as a “times sign.” An alternative in this situation is to use an asterisk (*). This is why you will occasionally see numbers written like this:

Scientific Notation Power-of-10 Notation The “times Sign”

Plain-text Exponents

Once in a while, you will have to express numbers in power-of-10 notation using plain, unformatted text. This is the case, for example, when transmitting information within the body of an e-mail message. Some calculators and computers use this system. An uppercase or lowercase letter E indicates that the quantity immediately following is a power of 10. The power-of-10 designator always includes a sign (plus or minus) unless it is zero. In this format, the above quantities are written like this:

Scientific Notation Power-of-10 Notation Plain-text Exponents

or like this:

Scientific Notation Power-of-10 Notation Plain-text Exponents

Another alternative is to use an asterisk to indicate multiplication, and the symbol ^ to indicate a superscript, so the expressions look like this:

Scientific Notation Power-of-10 Notation Plain-text Exponents

In all of the above examples, the numerical values, written out in fully expanded decimal form, look like this:

Scientific Notation Power-of-10 Notation Plain-text Exponents

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