Standard exponential form

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. 

None of these is in standard exponential form, because there are more or fewer than one digit to the left of the decimal point. (The 0 in 0.665 is not counted because it could just as well be omitted.) However, each of them has the same value as 6.65 X 105. 

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. 

Which of the following numbers are in standard exponential form ... 

All the digits in the coefficient of a number in standard exponential form are significant. 

The magnitudes of the answers are the same, just as the maguitudes of the origiual values are the same. The numbers of millimeters all look the same, but since we know where the values came from, we know how many significant digits each contains. We can solve the problem by using standard exponential form (a) 7.3 x 10 mm, (b) 7.30 x 10 mm,... 

An exponential number with a eoefficient greater than 10 or less than I can be changed to the standard exponential form by converting the coefficient to the standard form and adding the exponents ... 

An operator annihilates an electron in spin orbital i, while creates an electron in spin orbital a. The creation and annihilation operators satisfy the usual algebra [a, aj]+ = 6pg. Introducing the generic notation for excitation operators and noting that [r, iv] = 0, we may express the coupled-cluster wave function in the standard exponential form [35]... 

Numbers like this with trailing zeros are best written in scientific notation, where the number is written in the standard exponential form as N x 10", where N represents a number with a single non-zero digit to the left of the decimal point and n represents some integer. 

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

A, b moles of B, etc., as in the equation used to define K (Eq. 6-27 in this instance). It is also important to realize that the log term in Eq. 6-31 must be unitless. Although we usually write In K or log K, K here represents K / Q, where Q = 1 because it has the same form as K but with all components in their standard states. Since the units of K and Q are the same, log K is unitless. Similar considerations apply to expressions of K in exponential form. 

The acid AjH serves as our standard for comparison of all the others, so that log kt and log Kl are constants for a series of measurements therefore we can write Equation 3.53 (where C is a constant), which is equivalent to Equation 3.48 and, when written in exponential form, to Equation 3.47.61... 

Note The preceding illustrates the method used to prepare tabulated results of factors and en asizes the simpUdty of the procedure with a digital computer. This set of results represents a standard exponential function available in tabulated form in standard mathematical tables. See Prob. 8 at the end of diis dupter for a revested computer solution for a more conpUcated continuous-interest case.)... 

Scientists report numbers from literally astronomical to almost infinitesimal. In order to do so conveniently, we use scientific notation, also known as standard exponential notation. Scientific notation is a form of a number with a decimal coefficient times a power of 10. The following number is in scientific notation, with its parts identified ... 

Since some confusion has arisen in dielectric theory from careless definition of symbols, we have taken some care here to spell such matters out in some detail while adhering to standard usage. The electric field at any point will be described by the complex instantaneous field vector, (r, t) (in the low-frequency range where the magnetic effects are negligible and electrostatic conditions may be said to prevail) where. (r, t) = (r) ". We shall require the amplitude, (r) to be a complex vector so that the phase of (r, t) relative to other vectors may be written in exponential form. Thus... 

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