Powers of 10, for scientific notation

The easiest way to determine the appropriate power of 10 for scientific notation is to start with the number being represented and count the number of places the decimal point must be moved to obtain a number between 1 and 10. For example, for the number... 

The metric system is a decimal system, based on powers of 10. Table 2.5 is a list of the prefixes for the various powers of 10. Between scientific notation and the prefixes shown below, it is very simple to identify, name, read, and understand 36 decades of power of any given base or derived unit. 

Scientific notation uses exponents (powers of 10) for handling very large or very small numbers. A number in scientific notation consists of a number multiplied by a power of 10. The number is called the mantissa. In scientific notation, only one digit in the mantissa is to the left of the decimal place. The order of magnitude is expressed as a power of 10, and indicates how many places you had to move the decimal point so that only one digit remains to the left of the decimal point. 

A When adding and subtracting numbers in scientific notation, express the numbers to the same power of 10. For example,... 

Scientific notation simply expresses a number as a product of a number between 1 and 10 and the appropriate power of 10. For example, the number 93,000,000 can be expressed as... 

When a large or small number is written in standard scientific notation, the number is expressed as the product of a numher between 1 and 10, multiplied by the appropriate power of 10. For each of the following numbers, indicate what number between 1 and 10 would be appropriate when expressing the numbers in standard scientific notation. 

A number written in scientific notation has two parts a coefficient and a power of 10. For example, the number 2400 is written in scientific notation as 2.4 X 10. The coefficient, 2.4, is obtained by moving the decimal point to the left to give a number that is at least 1 but less than 10. Because we moved the decimal point three places to the left, the power of 10 is a positive 3, which is written as 10. When a number greater than 1 is converted to scientific notation, the power of 10 is positive. 

For example, to express 6,403,500,000 in scientific notation, first change the number to a decimal between 1 and 10, that is 6.4035. Now, multiply this decimal by a power of 10, determined by the number of placeholders the decimal was moved. This is a large number, so the power of 10 will be positive. 

To make working with such extreme numbers easier, chemists turn to scientific notation, which is a special kind of exponential notation. Exponential notation simply means writing a number in a way that includes exponents. In scientific notation, every number is written as the product of two numbers, a coefficient and a power of 10. In plain old exponential notation, a coefficient can be any value of a number multiplied by a power with a base of 10 (such as 10" ). But scientists have rules for coefficients in scientific notation. In scientific notation, the coefficient is always at least 1 and always less than 10. For example, the coefficient could be 7, 3.48, or 6.0001. 

Addition or subtraction gets easier when you express your numbers as coefficients of identical powers of 10. To wrestle your numbers into this form, you may need to use coefficients less than 1 or greater than 10. So scientific notation is a bit too strict for addition and subtraction, but exponential notation still serves you well. 

To multiply two numbers, put them both in standard scientific notation. Then multiply the two lefthand factors by ordinary multiplication, and multiply the two righthand factors (powers of 10) by the multiplication law for exponents — that is, by adding their exponents. 

We have discussed at length the usefulness of powers of 10 as part of scientific notation, but many practical problems involve the powers of other numbers. For example, the area of a circle involves the square of the radius, and the volume of a sphere involves the cube of the radius. Nearly every calculator yields the square of a number when you simply enter the number through the keyboard and then press the ke ihe S( uafe appears in the lighted display. 

Addition or subtraction. The values to be added or subtracted should all be expressed with the same units. If they are expressed in scientific notation, they should all be expressed with the same power of 10. The result should be rounded off so that it has only as many digits after the decimal point as the number with thefewest digits after the decimal. For example, consider the sum of the following weights ... 

One further point about significant figures Certain numbers, such as those obtained when counting objects, are exact and have an effectively infinite number of significant figures. For example, a week has exactly 7 days, not 6.9 or 7.0 or 7.1, and a foot has exactly 12 in., not 11.9 or 12.0 or 12.1. In addition, the power of 10 used in scientific notation is an exact number. That is, the number 103 is exact, but the number 1 X 103 has one significant figure. 

Scientific notation is a method of writing numbers in a specific format that is best explained by presenting an example. Suppose you want to write the number 2,376 in scientific notation. The first step is to rewrite the number putting the decimal immediately after the first digit, as in 2.376. Obviously 2,367 is not the same size number as 2.376, and so the 2.376 has to be multiplied by a power of 10. If 2.376 is multiplied by 1,000, we get the original number 2,376. Because 1,000 can be written as 103, the 2,376 can be written in scientific notation as 2.376 x 103. This format is called scientific notation. The goal is always for the number written in scientific notation to have the same value as the original number. 

Substitute simple integers for all numerical quantities, using powers of 10 (scientific notation) for very small and very large numbers. 

Scientists and engineers are frequently called upon to use very large or very small numbers for which ordinary decimal notation is either awkward or impossible. For example, to express Avogadro s number in decimal notation would require 21 zeros following the number 602. In scientific notation the number is written as a multiple of two numbers, the one number in decimal notation and the other expressed as a power of 10. Thus, Avogadro s number is written as 6.02 X 1Other examples are... 

Many times numbers used in science are very small or very large. Because these numbers are difficult to work with scientists use scientific notation. To write numbers in scientific notation, move the decimal point until only one non-zero digit remains on the left. Then count the number of places you moved the decimal point and use that number as a power of ten. For example, the average distance from the Sun to Mars is 227,800,000,000 m. In scientific notation, this distance is 2.278 x 10 m. Because you moved the decimal point to the left, the number is a positive power of ten. 

To represent a large number such as 20,500 in scientific notation, we must move the decimal point in such a way as to achieve a number between 1 and 10 and then multiply the result by a power of 10 to compensate for moving the decimal point. In this case, we must move the decimal point four places to the left. 

Scientific notation is a commonly used method for expressing large numbers. The number is written as a product of a decimal number and a power of 10. See the following question for an example of where this is useful. 

For each of the following numbers, if the number is rewritten in scientific notation, will the exponent of the power of 10 be positive, negative, or zero ... 

The prefixes in Table 2.4 are represented by the powers of 10 used in scientific, or exponential, notation for writing large and small numbers. For example, 10 = 10 X 10 X 10 = 1000. Appendix B reviews this notation. 

We often encounter very small and very large numbers in chemistry problems. For example, pesticide production in the world exceeds millions of tons, whereas pesticide residues that may harm animals or humans can have masses as small as nanograms. For either type of number, scientific notation is useful. Numbers written using scientific notation factor out all powers of ten and write them separately. Thus the number 54,000 is written as 5.4 x lO. This notation is equivalent to 5.4 X 10,000, which clearly is 54,000. Small numbers can also be written in scientific notation using negative powers of ten, because 10 is identical to 1/10. The number 0.000042 is 4.2 x 10 in scientific notation. 

Scientists often use numbers that are very large or very small in measurements. For example, the Earth s age is estimated to be about 4,500,000 (4.5 billion) years. Numbers like these are bulky to write, so to make them more compact scientists use powers of 10. Writing a number as the product of a number between 1 and 10 multiplied by 10 raised to some power is called scientific notation. 

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