real values

Examples of real values in the following topics:

• Imaginary Numbers

  • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
  • What does it mean, then, if the value under the radical is negative, such as in $\displaystyle \sqrt{-1}$?
  • There is no real value such that when multiplied by itself it results in a negative value.
  • This means that there is no real value of $x$ that would make $x^2 =-1$ a true statement.
  • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.

• Calculating Real GDP

  • Real GDP growth is the value of all goods produced in a given year; nominal GDP is value of all the goods taking price changes into account.
  • The real GDP is the total value of all of the final goods and services that an economy produces during a given year, accounting for inflation .
  • In economics, real value is not influenced by changes in price, it is only impacted by changes in quantity.
  • Real values measure the purchasing power net of any price changes over time.
  • Real GDP accounts for inflation and deflation.

• Ideal vs. Real Culture

  • When we talk about American values, we often have in mind a set of ideal values.
  • Along with every value system comes exceptions to those values.
  • With these exceptions, real values emerge.
  • Whereas we might refer to ideal values when listing American values (or even our own values), the values that we uphold in daily life tend to be real values.
  • In ideal culture, marriage is forever, but in real culture, many marriages end in divorce.

• Absolute Value

  • Absolute value can be thought of as the distance of a real number from zero.
  • In mathematics, the absolute value (sometimes called the modulus) of a real number $a$ is denoted $\left | a \right |$.
  • For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
  • When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.
  • Other names for absolute value include "numerical value," "modulus," and "magnitude."

• Other Considerations in Capital Budgeting

  • The real option creates economic value by generating future decision rights for management.
  • A key feature is that the real option creates economic value by generating future decision rights - specifically, by offering management the flexibility to act upon new information such that the upside economic potential is retained while the downside losses are contained .
  • Another value-creating aspect of real options can be found in abandonment.
  • Real investments are often made not only for immediate cash flows from the project, but also for the economic value derived from subsequent investment opportunities.
  • Projects with real options can be evaluated using a range of possible profits.

• Visualizing Domain and Range

  • In other words, two different values of $x$ can have the same $y$-value, but each $y$-value must be joined with a distinct $x$-value.  
  • Both graphs include all real numbers $x$ as input values, since both graphs continue to the left (negative values) and to the right (positive values) for $x$ (inputs).  
  • The curves continue to infinity in both directions; therefore, we say the domain for  both graphs is the set of all real numbers, notated as: $\mathbb{R}$.
  • The graph of $f(x)=x^2$ (red) has the same domain (input values) as the graph of $f(x)=-\frac{1}{12}x^3$ (blue) since all real numbers can be input values.  
  • The range of the blue graph is all real numbers, $\mathbb{R}$.

• Differences Between Real and Nominal Rates

  • The real rate is the nominal rate minus inflation.
  • In the case of a loan, it is this real interest that the lender receives as income.
  • A lender would have no net benefit from such a loan because inflation fully diminishes the value of the loan's profit.
  • Where r is the real rate, i is the inflation rate, and R is the nominal rate.
  • The real rate can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate: 1 + i = (1+r) (1+E(r)), where i = nominal interest rate; r = real interest rate; E(r) = expected inflation rate.

• Zeros of Polynomial Functions with Real Coefficients

  • A root, or zero, of a polynomial function is a value that can be plugged into the function and yield $0$.
  • The zero of a function, $f(x)$, refers to the value or values of $x$ that will result in the function equaling zero, $f(x)=0$.
  • This section specifically deals with polynomials that have real coefficients.
  • (An example of a non-real number would be $\sqrt -1$.)
  • Even though all polynomials have roots, not all roots are real numbers.

• The GDP Deflator

  • Nominal GDP, or unadjusted GDP, is the market value of all final goods produced in a geographical region, usually a country.
  • That market value depends on the quantities of goods and services produced and their respective prices.
  • In other words, real GDP is nominal GDP adjusted for inflation.
  • Real GDP reflects changes in real production.
  • It is calculated by dividing nominal GDP by real GDP and multiplying by 100.

• Bibliography

  • Elementary differential equations and boundary value problems.
  • Real analysis.