Numbers scientific 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. 

Handling Numbers Scientific notation is used to express large and small numbers, and each number in a measurement must indicate the meaningful digits, called significant figures. 

Measured Quantity Standard Number Scientific Notation ... 

In general, any ambiguity concerning the number of significant figures in a measurement can be resolved by using exponential notation (often referred to as scientific notation ), discussed in Appendix 3. 

Numbers such as these are very awkward to work with. For example, neither of the numbers just written could be entered directly on a calculator. Operations involving very large or very small numbers can be simplified by using exponential (scientific) notation. To express a number in exponential notation, write it in the form... 

In scientific notation, a number is written as A x 10 7. Here A is a decimal number with one nonzero digit in front of the decimal point and a is a whole number. For example, 333 is written 3.33 X 102 in scientific notation, because 102= 10x 10 = 100 ... 

To multiply numbers in scientific notation, multiply the decimal parts of the numbers and add the powers of 10 ... 

When raising a number in scientific notation to a particular power raise the decimal part of the number to the power and multiply the power of 10 by the power ... 

As usual, Feynman was right. His little particles captures an essential fact about atoms. They are tiny—so tiny that a teaspoon of water contains about 500,000,000,000,000,000,000,000 of them. Handling numbers this big is awkward. Try dividing it by 63, for example. To accommodate the very large numbers encountered in counting atoms and the very small ones needed to measure them, chemists use the scientific notation system. 

Scientific notation uses exponents to express numbers. The number 1,000, for instance, is equal to 10 x 10 x 10, or 10. The number of zeros following the 1 in 1,000 is 3, the same as the exponent in scientific notation. Similarly, 10,000, with 4 zeros, would be 10 , and so on. The same rules apply to numbers that are not even multiples of 10. For example, the number 1,360 is 1.36 x 10. And the number of atoms in a spoonful of water becomes an easy-to-write 5 X 10. ... 

Scientific notation is also useful for representing very small numbers. The number 0.1 would be 1/10 or 10". The radius of an atom of aluminum is 0.000000000143 meters. Using scientific notation, we could write this distance more compactly as 1.43 x 10 °. Atoms are, indeed, very little particles. ... 

The abbreviation log stands for logarithm. In mathematics, a logarithm is the power (also called an exponent) to which a number (called the base) has to be raised to get a particular number. In other words, it is the number of times the base (this is the mathematical base, not a chemical base) must be multiplied times itself to get a particular number. For example, if the base number is 10 and 1,000 is the number trying to be reached, the logarithm is 3 because 10 x 10 x 10 equals 1,000. Another way to look at this is to put the number 1,000 into scientific notation ... 

When scientists write numbers in exponential form, they prefer to write them so that the coefficient has one and only one digit to the left of the decimal point. That notation is called standard exponential form or scientific notation. 

Note (2) It is more difficult to find K in scientific notation because most calculators cannot handle numbers this big. So, use what you know about exponents to solve for K ... 

B This is easier to visualize if the numbers are not in scientific notation. 

Most people find that writing 300000000 ms-1 is a bit long winded. Some people do not like writing simple factors such as G for giga, and prefer so-called scientific notation. In this style, we write a number followed by a factor expressed as ten raised to an appropriate power. The number above would be 3.0 x 108 ms-1. 

A factor is simply shorthand, and is dispensable. We could have dispensed with the factor and written the number differently, saying energy = 12000 Jmol-1. This same energy in scientific notation would be 12 x 103 J mol-1. But units are not dispensable. 

Round the following numbers to the number of significant figures indicated and express in scientific notation. 

Chemistry is full of calculations. Our basic goal is to help you develop the knowledge and strategies you need to solve these problems. In this chapter, you will review the Metric system and basic problem solving techniques, such as the Unit Conversion Method. Your textbook or instructor may call this problem solving method by a different name, such as the Factor-Label Method and Dimensional Analysis. Check with your instructor or textbook as to for which SI (Metric) prefixes and SI-English relationships will you be responsible. Finally, be familiar with the operation of your calculator. (A scientific calculator will be the best for chemistry purposes.) Be sure that you can correctly enter a number in scientific notation. It would also help if you set your calculator to display in scientific notation. Refer to your calculator s manual for information about your specific brand and model. Chemistry is not a spectator sport, so you will need to Practice, Practice, Practice. 

Any zero to the right of nonzero digits and to the left of a decimal point and not covered by rule 2 may or may not be significant, depending on whether the zero is a placeholder or actually part of the measurement Such a number should be expressed in scientific notation to avoid any confusion. 

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. 

Since the decimal point was moved nine places to the left, the power of 10 is nine, and the number written in scientific notation is 6.4035 x 109. Note that even though 64.035 x 108 is another form of the same number, this is NOT scientific notation, since the decimal number is not between one and ten. As an example of a very small number, consider changing 0.000006007 to a number expressed in scientific notation. Write the num-... 

Scientific notation is a shorthand way to express very large or very small numbers. The notation expresses a number as a decimal (between one and ten), multiplied by an appropriate power of ten. 

Rule 1 To multiply two numbers in scientific notation, add the exponents. 

Rule 3 To add or subtract numbers in scientific notation, first convert the numbers so they have the same exponent. Each number should have the same exponent as the number with the greatest power of 10. Once the numbers are all expressed to the same power of 10, the power of 10 is neither added nor subtracted in the calculation. 

---