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An ordinary annuity refers to a series of equal payments made at the end of consecutive periods over a fixed length of time. When it comes to saving for retirement or other long-term goals, ordinary annuities can be very beneficial because they allow individuals to accumulate a large sum of money over time by making regular, manageable contributions.

To calculate the future value of an ordinary annuity, we use a specific formula that takes into account the amount of each payment, the rate of compound interest, and the number of periods the payments are made. The exercise provided illustrates how the future value of an ordinary annuity can be computed given a $4000 annual payment, 6.5% interest rate, and a 40-year period.

Understanding the principles behind ordinary annuities helps in financial planning, especially when considering the benefits of long-term, regular investments and the impact of interest over time.

Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated solely on the principal amount, compound interest allows for the interest to 'compound' — meaning the interest earns interest over time.

In the given exercise, the annual compounding makes a significant difference over 40 years, as each year's accumulated interest is folded into the principal for the calculation of the next year's interest. This powerful effect of compounding results in a much larger amount than would be calculated using simple interest. The formula used in Step 1 of the solution leverages this concept to find the future value of the ordinary annuity.

Present value is a financial concept that represents the current value of a sum of money to be received in the future, discounted to reflect the time value of money. In essence, it answers the question: 'How much is future money worth today?'

The concept of present value is tied closely to compound interest. While compound interest grows a present sum into a larger future sum, present value works in reverse, shrinking a future sum into a present equivalent based on the expected rates of return or interest rates. While the concept of present value isn't directly addressed in the exercise steps, it underlies the processes used to compute the future values and interest, as these calculations essentially reverse the present value logic.

Financial mathematics is a branch of applied mathematics that analyzes and solves problems related to financial markets. It encompasses concepts like ordinary annuity, compound interest, and present value, among others. These elements are crucial for making informed financial decisions, whether by individuals or professionals in finance.

In the given exercise, the application of financial mathematics is evident through the formulas and computations needed to solve for the future value of an annuity and the interest earned over time. Mastery of these financial mathematics principles enables individuals to better understand investment growth, interest accrual, and the overall time value of money, providing a strong foundation for personal financial management and investment analysis.