Chapter 1 is a comprehensive review of math fundamentals including algebra,equation solving, and functions. The exposition of each topic is brief to make foreasy reading

so a formula is like a dead equation?

Axiom (Dynamics). There is a distinguished quantum observable, the Hamil-tonian H. Time evolution of states |ψ(t)〉 ∈ H is given by the Schr ̈odingerequationi~ ddt |ψ(t)〉 = H|ψ(t)〉 (1.1)

components

mathematically

where R has units of 8.314J/mol-K

algebra creates a useful equation

I'd say an equation is anything with an equals sign in it; a formula is an equation of the form A= stuffA= stuffA={\rm\ stuff} where AAA does not appear among the stuff on the right side.

I think that over time the distinction is lost. My math teacher, 35 years ago stated "formulas are used in chemistry, in math we have equations". To this day, the word 'formula' in math seems wrong, but I'd accept it's used commonly.

An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.

The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.

C7H16(l)+O2(g)→CO2(g)+H2O(g)(3.1.4)(3.1.4)C7H16(l)+O2(g)→CO2(g)+H2O(g) C_7H_{16} (l) + O_2 (g) \rightarrow CO_2 (g) + H_2O (g) \label{3.1.3} The complete combustion of any hydrocarbon with sufficient oxygen always yields carbon dioxide and water. Figure 3.1.23.1.2\PageIndex{2}: An Example of a Combustion Reaction. The wax in a candle is a high-molecular-mass hydrocarbon, which produces gaseous carbon dioxide and water vapor in a combustion reaction (see Equation 3.1.43.1.4\ref{3.1.3}). Equation 3.1.43.1.4\ref{3.1.3} is not balanced: the numbers of each type of atom on the reactant side of the equation (7 carbon atoms, 16 hydrogen atoms, and 2 oxygen atoms) is not the same as the numbers of each type of atom on the product side (1 carbon atom, 2 hydrogen atoms, and 3 oxygen atoms). Consequently, the coefficients of the reactants and products must be adjusted to give the same numbers of atoms of each type on both sides of the equation. Because the identities of the reactants and products are fixed, the equation cannot be balanced by changing the subscripts of the reactants or the products. To do so would change the chemical identity of the species being described, as illustrated in Figure 3.1.33.1.3\PageIndex{3}. Figure 3.1.33.1.3\PageIndex{3}: Balancing Equations. You cannot change subscripts in a chemical formula to balance a chemical equation; you can change only the coefficients. Changing subscripts changes the ratios of atoms in the molecule and the resulting chemical properties. For example, water (H2O) and hydrogen peroxide (H2O2) are chemically distinct substances. H2O2 decomposes to H2O and O2 gas when it comes in contact with the metal platinum, whereas no such reaction occurs between water and platinum. The simplest and most generally useful method for balancing chemical equations is “inspection,” better known as trial and error. The following is an efficient approach to balancing a chemical equation using this method.

This, of course, leaves us none the wiser as to how to model velocity, as the equation of exchange is nothing more than an identity. MV=PQ just says that the money flow of expenditures is equal to the market value of what those expenditures buy, which is true by definition. The left and right sides are two ways of saying the same thing; it’s a form of double-entry accounting where each transaction is simultaneously recorded on both sides of the equation. Whether an effect should be recorded in M, V, P, or Q is, ultimately, arbitrary. To transform the identity into a tool with predictive potency, we need to make a series of assumptions about each of the variables. For example, monetarists assume M is determined exogenously, V is constant, and Q is independent of M and use the equation to demonstrate how increases in the money supply increase P (i.e. cause inflation).

The first practical problem with velocity is that it’s frequently employed as a catch-all to make the two sides of the equation of exchange balance. It often simply captures the error in our estimation of the other variables in the model.

Two people can have one conversation. Three people have four unique conversation groups (three different two-person conversations and a fourth conversation between all three as a group). Five people have 26. Twenty people have 1,048,554.

Self-Acceptance + Self Assurance = Confidence

cases which are never equal and thus altogether unequal. Every concept arises from the equation of unequal things.