Examples of difference in the following topics:
-
- Forward difference: $\Delta F(P) = F(P + \Delta P) - F(P)$;
- Central difference: $\delta F(P) = F(P + P) - F(P - P)$
- Backward difference: $\nabla F(P) = F(P) - F(P - \Delta P)$.
- The function difference divided by the point difference is known as the difference quotient, attributed to Isaac Newton.
- Relate the difference quotient in algebra to the derivative in calculus
-
- An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
- An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
- For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
- The behavior of the arithmetic sequence depends on the common difference $d$.
-
- When a quadratic is a difference of squares, there is a helpful formula for factoring it.
- When we recall this fact in the reverse direction, it is called the difference of squares formula.
- By rewriting the equation as $x^2-a^2=0$ and factoring the difference of squares, you would obtain $(x-a)(x+a)=0.$ Thus there are two solutions, where $x-a=0$ (so $x=a$) and where $x+a=0$, (so $x=-a$).
- This latter equation has no solutions, since $4x^2$ is always greater than or equal to $0.$ However, the first equation $4x^2-3=0$ can be factored again as the difference of squares, if we consider $3$ as the square of $\sqrt3$.
- Evaluate whether a quadratic equation is a difference of squares and factor it accordingly if it is.
-
- The difference between 7 and 5 is 2.
- The difference between 7 and 9 is also 2.
- Since this difference is constant, and this is the first set of differences, the sequence is given by a first-degree (linear) polynomial.
- The difference between -7 and 4 is 11, the difference between -26 and -7 is -19.
- Because this term is not a polynomial, taking differences will never result in a constant difference.
-
- The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities.
- Similarly, the logarithm of the ratio of two quantities is the difference of the logarithms.
-
- Variables are useful in mathematics for many reasons, and can be used to denote different types of arbitrary or unknown numbers.
- Variables can be used to represent different types of numbers.
- It is common that many variables appear in the same mathematical formula, and they may play different roles.
- To distinguish between the different variables, $x$ is called an unknown, and the variables that are multiplied by $x$ are called coefficients.
-
- These techniques can be used in calculating sums, differences and products of company information such as sodas that come in three different flavors: apple, orange, and strawberry and two different packaging: bottle and can.
- You cannot add two matrices that have different dimensions.
- Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions.
-
- A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points.
- Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from GPS transmitters.
- The can also be characterized as the difference in arrival times of synchronized signals between the desired point and known points.
- In the case where a ship, or other object to be located, only knows the difference in distances between itself and two known points, the curve of possible locations is a hyperbola.
- So if we call this difference in distances 2a, the hyperbola will have vertices separated by the same distance 2a, and the foci of the hyperbola will be the two known points.
-
- We want the set of all points that have the same difference between the distances to these points.
- Then the difference of distances between $P$ and the two focal points is:
- With this value for the difference of distances, we can choose any point $(x,y)$ on the hyperbola and construct an equation by use of the distance formula.
- There is another common form of hyperbola equation that, at first glance, looks very different: $y = \frac{1}{x}$ or $xy = 1$.
- To prove that it is the same as the standard hyperbola, you can check for yourself that it has two focal points and that all points have the same difference of distances.
-
- An equation states that two expressions are equal, while an inequality relates two different values.
- An inequality is a relation that holds between two values when they are different.