Enclosure symbols

TABLE 9.1 Dirac Enclosure Symbols for Different Types of Mathematical Object, with Matrix-Algebraic Examples... 

The essence of Dirac s notation is to distinguish different types of mathematical objects by enclosures around symbols, rather than differences in the symbols themselves. We can associate different enclosure brackets (or portions thereof) with scalars, vectors (ordinary or adjoint), and matrices (operators) as shown in Table 9.1. 

Parentheses and division bars (fraction bars) are both symbols of enclosure. Whenever there are symbols of enclosure, execute any arithmetic operations inside the symbol of enclosure first. The general rule is to work from the innermost symbol of enclosure to the outermost, multiplying and dividing then adding and subtracting. Consider this example, which involves both kinds of symbols of enclosure and order of operations. 

Here we come to a very common error. Many students are tempted to divide 30 by 3 first, and then multiply by 5. But the division bar is a symbol of enclosure, so multiply 3 times 5 first and then divide 30 by 15. 

Remember to follow the rules for symbols of enclosure. The division bar is a symbol of enclosure, so do the operations in the numerator and in the denominator before performing the division. By the way, did you notice you can easily solve this problem without a calculator Try regrouping the mantissas separately from the powers of 10. 

Apply these rules according to the algebraic hierarchy of calculations (consider symbols of enclosure first, then multiplication/division operations and lastly addition/subtraction operations). 

A division bar is a symbol of enclosure do the subtraction first, keeping three decimal places ... 

The enclosure of the symbol for oxygen implies the liberation of highly reactive oxygen. 

Activity is given in the same units as concentration—molarity, mole fraction, and so forth. In (2-9) the concentration C usually is expressed either in moles per liter of solution (molarity M) or in moles per kilogram of solvent (molality m). In dilute aqueous solutions the molarity and molality are nearly equal in nonaqueous solutions molarity is usually larger than molality, since the density of the solvent is usually less than unity. Analytical chemists ordinarily find it more convenient to express concentration in molarity, even though it varies slightly with temperature. The analytical concentration is represented by the symbol C, to indicate the moles of solute added per liter of solution. Analytical concentration should be distinguished from the equilibrium concentration, which is indicated by enclosure in square brackets. 

Airborne radiation can appear in any room, enclosure, or operating area in which radioactive materials exist in concentrations in excess of the amounts specified in Table 1 of Appendix B to 10 CFR 20. Post a sign or signs bearing the radiation caution symbol and the word CAUTION in areas where exposure could occur. 

The symbol for the fuel cell and electrolysis cell is derived from the battery symbol the longer and shorter lines represent, respectively, the cathode and anode, and the dashed line represents the electrolyte. An arrow drawn in the direction of positive current flow points toward an electrolyte with negative charge carriers, as in the manner of the transistor symbol. Galvanic and electrolytic cells are distinguished by the location of the positive terminal a positive terminal at the cathode indicates a galvanic cell, while a positive terminal at the anode indicates an electrolytic cell. The outer box represents the system enclosure, which may or may not be open. The values of potential and overpotential are consistent with Table 2. 

Storage units. Storage enclosures, such as refrigerators, freezers, or even incubators used to store or hold biohazardous agents, should be identified with a sign or label containing the biohazard symbol and hazard identity. 

Names of independent quantities are separated hy colons alternative names for a single quantity are separated hy semi-colons. Enclosure of the name of a quantity in parentheses indicates that there is an alternative symbol of equal standing enclosure in square hrackets indicates that there is another symbol which is preferred. An asterisk against the name of a quantity indicates that the specified symbol is always used with a svbscript when denoting that quantity.)... 

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