Numerical Mathematical Operations

Round the following numbers to four significant digits (b) 78955 (d) 46.4535 

We are frequently required to carry out numerical operations on numbers. The first such operations involve pairs of numbers. 

The elementary mathematical operations are addition, subtraction, multiplication, and division. Some rules for operating on numbers with sign can be simply stated  

The product of two factors of the same sign is positive, and the product of two factors of different signs is negative. 

---

In Subheading 2.3. the important class of vectors with continuous-valued components is described. A number of issues arise in this case. Importantly, since the objects of concern here are vectors, the mathematical operations employed are those applied to vectors such as addition, multiplication by a scalar, and formation of inner products. While distances between vectors are used in similarity studies, inner products are the most prevalent type of terms found in MSA. Such similarities, usually associated with the names Carbo and Hodgkin, are computed as ratios, where the inner product term in the numerator is normalized by a term in the denominator that is some form of mean (e.g., geometric or arithmetic) of the norms of the two vectors. 

Probability bounds analysis combines p-boxes together in mathematical operations such as addition, subtraction, multiplication, and division. This is an alternative to what is usually done with Monte Carlo simulations, which usually evaluate a risk expression in one fell swoop in each iteration. In probability bounds analysis, a complex calculation is decomposed into its constituent arithmetic operations, which are computed separately to build up the final answer. The actual calculations needed to effect these operations with p-boxes are straightforward and elementary. This is not to say, however, that they are the kinds of calculations one would want to do by hand. In aggregate, they will often be cumbersome and should generally be done on computer. But it may be helpful to the reader to step through a numerical example just to see the nature of the calculation. 

The fourth module is the graphics waveform postprocessor. It has the feel of an oscilloscope, translating SPICE s numerical data output into waveforms. Many different mathematical operations can be performed, such as integration, FFTs, etc. allowing users to get the most out of SPICE. 

The special branch of mathematics known as numerical analysis has assumed an added importance with the extensive use of digital computers. Since these calculators perform only the fundamental operations of arithmetic, it is necessary that all other mathematical operations be reduced to these terms. From a superficial viewpoint it might be concluded that such operations as differentiation and integration are inherently better suited to analog computers. This is not necessarily true, however, and depends upon the requirements of the particular problem at hand. 

Mathematical operations have specific rules for the use of mathematical symbols with SI units. A space or a half-high dot represents the multiplication of units a negative exponent, horizontal line, or slash represents the division of units, and if these mathematical symbols appear in the same line, parentheses must differentiate them. The percent sign (%) denotes the number 0.01 or 1/100, so that 1%= 0.01, 30% = 0.30, and so forth. Arabic numerals with the appropriate SI or recognized unit indicate the values of quantities. Commas are not used to separate numbers into groups of three. If more than four digits appear on either side of the decimal point, a space Table 3. Prefixes. separates the groups of three. 

Numbers, unit symbols, and names have set rules for mixing and differentiation for clarity of text and mathematical operations. These include a space between a numerical value and its unit symbol, indicating clearly the number a symbol belongs to in a given mathematical calculation, and no mixing of unit symbols and names nor making calculations on unit names. Different symbols represent values and units and the unit symbol should follow the value symbol separated by a slash. SI requires the use of standardized mathematical symbols and the explicit writing of a quotient quantity. 

If you refer to the section on Accuracy you will notice in the first example given on relative error calculation that the numerator was only 2 SF (0.10) after the subtraction step. Therefore, the denominator was rounded off to 2 SF (12) and the answer was expressed as 2 SF. The division by 3 to get the mean does not limit the SF to one because the 3 is part of a mathematical operation and not an experimental value. Some sources specify rounding off during a calculation while others say it should be done only at the end. In the relative error calculation, the answer is changed by 0.02% if rounding off is done at the end as shown below ... 

One of the most useful features of spreadsheets is the ease with which repetitive calculations can be done by copying the formula in one cell to others. A formula in a cell is a mathematical operation that can utilize values contained in other cells, as shown by the formula content of the active cell B14. This formula is displayed in the information bar at the top of the spreadsheet of Fig. 1 and is the equation for the Morse potential function, with the cell addresses of the constants Dg, /3, and the variable r entered instead of numerical values. By using the Edit menu (or the right mouse button), the formula operation of B14 can be Copied and then Pasted into cells B15, B16.B73, thus giv-... 

It is possible, however, to substitute for any mathematical operation such as differentiation or integration a series of arithmetic operations that yield approximately the same result. The arithmetic operations are usually simple but numerous and repetitious and so are ideally suited to computers. 

In this chapter we will encounter a number of standard mathematical operations that are conveniently performed and/or illustrated on a spreadsheet. We start with a brief description of the logic underlying the Goal Seek and Solver methods of Excel. Then we consider two methods often encountered in spectroscopy, viz. signal averaging and lock-in amplification. Subsequently the focus shifts toward numerical methods, such as peak fitting, integration, differentiation, and interpolation, some of which we have already encountered in one form or another in the context of least squares analysis and/or Fourier transformation. Finally we describe some matrix operations that are easy to perform with Excel. 

Powerl first reads the numerical values of all cells, and stores them in an array. The mathematical operations are then performed on these stored data, so that there is no ambiguity involved. Consequently, Powerl yields correct results for either column. However, Powerl may fail when the selected block is too large (which, depending on the amount of available memory, and the version of Excel used, typically will happen only when the array has more than a few thousand elements). In that case, use Special Copy Values to duplicate the numerical values of the array elements in another block, then use a macro such as Power that does not suffer from size limitations. 

Spreadsheets are very powerful and convenient computational tools to illustrate mathematical relationships, and to solve numerical problems, as demonstrated in this book within the context of analytical chemistry. The original spreadsheets were poorly suited to perform some types of mathematical operations, such as iterations. Fortunately, the open structure of modern Excel allows the user to introduce extra features, by incorporating additional programs that accommodate particular needs. Macros are the most convenient and user-friendly way to give the spreadsheet such... 

Now one can realise that the mosaic of theoretical magnetochemistry has roots in the special theory of relativity, the theory of tensor operators, group theory, quantum mechanics, quantum chemistry, statistical thermodynamics, numerical mathematics, etc. 

The antecedent and consequent of a rule consist of two parts an object and its value, which are linked by an operator. The operator identifies an object and assigns a value. Operators, such as is, are, is not and are not, are usually used to assign a symbolic value to a linguistic term. Mathematical operators can also be used to define an object as numerical and assign it a numerical value. For example,... 

Numerical methods are a family of mathematical techniques for solving complicated problems approximately by the repetition of the elementary mathematical operations (+, —, x, H-). In the past, numerical solution of a PDE was extremely time consuming, requiring many man-hours of tedious calculation. However the utility of numerical methods has greatly increased since the advent of programmable computers. In the field of electrochemistry, two main techniques are used for simulation purposes the finite difference method and the finite element method, though the former is by far the more popular and will be used exclusively throughout this book. 

Once the type of representation has been selected, the similarity function or coefficient must be evaluated with respect to the chosen representation. Most of the similarity functions in use today are computed as ratios. The numerator is generally associated with some measure of the common features of the molecules being compared, while the denominator is associated with some measure of all of the features of the molecules. The mathematical forms of the similarity functions are quite similar, although evaluation of the expressions may require different types of mathematical operations. More specifically, the expressions set- and graph-based representations can be grouped together as can vector- and function-based representations. 

SI units are divided into three classes—base, supplementary, and derived—and these are given in Tables I-III. No decision has been taken regarding the classification of the supplementary units, i.e., whether they should be regarded as base or derived units. The derived units are formed by simple mathematical operations on the base or supplementary units without the introduction of any numerical factors. Some of the derived units have special names and symbols and these are included in Table III. 

---