Exponential notation,

In exponential notation, a number is represented as a value raised to a power of ten. The decimal point can be located anywhere within the number as long as the power of ten is correct. In scientific notation, the decimal point is always located between the first and second digit — and the first digit must be a number other than zero. 

Suppose, for example, that you have an object that s 0.00125 meters in length. You can express that munber in a variety of exponential forms  

All these forms are mathematically correct as numbers expressed in exponential notation. In scientific notation, the decimal point is placed so that there s one digit other than zero to the left of the decimal point. In the preceding example, the number e]q ressed in scientific notation is 1.25 x 10 m. Most scientists automatically express numbers in scientific notation. 

Here are some positive and negative powers of ten and the numbers they represent  

By analogous reasoning, m must also be even. But as we have seen, n and m cannot both be even. Therefore, the assumption that /2 is rational must be false. The decimal expansion V2 = 1.414 213 562 373... is nonterminating and nonperiodic. 

000 000 910 938 188 kg. Counting 31 places to the right, we write this number much more reasonably as 9.10938 x 10 kg, accurate to six significant figures. 

Most htde kids know the sequence of doubled numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 — These are, of course, successive powers of 2. Since the internal workings of computers is based on the binary number system, a memory capacity of 2 ° = 1024 bytes is conventionally called 1 kilobyte and a CPU speed of 1024 hertz is 1 kilohertz. Likewise, multiples of 2 ° = 1048576 1.05 x 10 are called megabytes (MB) and 

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The number of significant figures is the number of digits shown when a quantity is expressed in exponential 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... 

A major advantage of exponential notation is that it simplifies the processes of multiplication and division. To multiply, add exponents ... 

It often happens that multiplication or division yields an answer that is not in standard exponential notation. For example,... 

The product is not in standard exponential notation because the coefficient, 30, does not lie between 1 and 10. To correct this situation, rewrite the coefficient as 3.0 X 101 and then add exponents ... 

On all scientific calculators it is possible to enter numbers in exponential notation. The method used depends on the brand of calculator. Most often, it involves using a key labeled [exp), [ ee 1, or [eex]. Check your instruction manual for the procedure to be followed. To make sure you understand it, try entering the following numbers ... 

Experimental yield The amount of product actually obtained in a reaction, 65 Exponential notation, 643-645 Extensive property A property of a... 

Appendix 3 contains a mathematical review touching on just about all the mathematics you need for general chemistry. Exponential notation and logarithms (natural and base 10) are emphasized. 

To be sure that you know how many significant digits there are in such a number, you can report the number in standard exponential notation. The population of New York would be 8 X 10h people, and the bank account would be 8.000000 x 106 dollars. All digits in standard exponential form arc significant. 

Explain why every 0 in the coefficient is significant in a number properly expressed in standard exponential notation. 

Express, in standard exponential notation, the number of ( ) cubic centimeters in 1.0L... 

Standard exponential notation exponential notation with the coefficient having a value of 1 or more but less than 10. 

Tables C. 1-C.4 provide conversion factors from a.u. to SI units and a variety of practical (thermochemical, crystallographic, spectroscopic) non-SI units in common usage. Numerical values are quoted to six-digit precision (though many are known to higher accuracy) in an abbreviated exponential notation, whereby 6.022 14(23) means 6.022 14 x 1023. In this book we follow a current tendency of the quantum chemical literature by expressing relative energies in thermochemical units (kcal mol-1), structural parameters in crystallographic Angstrom units (A), vibrational frequencies in common spectroscopic units (cm-1), and so forth. These choices, although inconsistent according to SI orthodoxy, seem better able to serve effective communication between theoreticians and experimentalists.

The pH is defined as the logarithm of the reciprocal of the hydrogen ion concentration. The pH value of topical dosage forms is adjusted for various reasons including minimization of discomfort, maintenance of chemical stability, and improvement of therapeutic response. The values of hydronium ion concentration are very small and are therefore expressed in exponential notations as pH. 

Similarly, the subtraction of the exponential notation may be performed by changing the expressions to forms having the same common power of 10 and then subtracting the coefficients. 

Another way to determine the number of significant figures in a number is to express it in scientific (exponential) notation. The number of digits shown is the number of significant figures. For example, 2.305 X 10 5 would contain four significant figures. 

N is a numeric field. The contents are numeric valims that have a field length of 13. Numeric values are represented by a maximum of 7 significant figures when the field is shown in exponential notation. Any of the following notations are also acceptable ... 

Crunching numbers in scientific and exponential notation Telling the difference between accuracy and precision Doing math with significant figures... 

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. 

A major benefit of presenting numbers in scientific notation is that it simplifies common arithmetic operations. The simplifying abilities of scientific notation cire most evident in multiplication and division. (As we note in the next section, addition and subtraction benefit from exponential notation but not necesscirily from strict scientific notation.)... 

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 add two numbers easily by using exponential notation, first express each number as a coefficient and a power of 10, making sure that 10 is raised to the Scime exponent in each number. Then add the coefficients. To subtract numbers in exponential notation, follow the same steps but subtract the coefficients. 

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