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**Increment|Definition & Meaning**

## Definition

**Increment** means the **amount **or** value** by which something grows. In math, if the value of a **variable increases** by some amount, then that amount is called an increment.

The term increment describes the **positive changes **in variable values. Simply put, it represents the arithmetic operation of addition. There is an **original value** (the first addend) and the value **by which it increases** (the second addend). The latter is the increment, and the final result is the sum of the two.

**original value + increment = new larger value**

Consider that you have two apples. An increment of one apple would then give you three apples, as shown below:

Usually, you will see it used to represent **small increases**. **Gain **is another term that means the same thing, but most people use it to show a **large amount **of change (like the gain of guitar amplifiers).

The delta symbol “$\Delta$” denotes a change in value and is therefore used to represent increments. For example, if the variable is **x**, then $\boldsymbol\Delta$**x** represents the increment such that the result is **x + **$\boldsymbol\Delta$**x**.

## Decrements

**Decrements **are the opposite of increments: they show **small negative changes** in values. In other words, a decrement represents the value **subtracted **from some original amount.

**original value – decrement = new smaller value**

Consider the same apple example from before:

Thus, a change in some variable’s value can either be an increment or a decrement. A decrement is represented with **x – **$\boldsymbol\Delta$**x**.

**Some Practical Examples Using Increments**

Since increments are small additions to some original value, let us consider **simple counting**. When we count 1, 2, 3, we are simply **incrementing by 1 **for every next entry. Therefore, if we start at 1 (first entry), the next one is 1 + 1 = 2 (second entry). The number after 2 is 2 + 1 = 3 (third entry), and so on.

Now consider the set of positive even numbers {0, 2, 4, 6, 8, …}. Here, entry number x$_\mathsf{i+1}$ = x$_\mathsf{i}$ + $\Delta$ x where $\Delta$x = 2. Thus, we have an increment of 2 for each subsequent entry in this sequence.

Now, suppose you want to buy tickets to a fair. The **base price** for a single person is \$15. However, there is a **family offer for up to five members**, such that for each extra member, you only have to pay **\$5**. We illustrate this below:

Thus, each additional family member **saves** **\$10** by using the family coupon. If each person bought a separate ticket, the cost of five tickets would be **5 x \$15 = \$75**. With the **family offer on five members**, everybody gets to enjoy the fair with just **\$35**, **saving** **\$75 – \$35 = \$40**!

## Increments in Functions

Consider a straight line with a slope equal to m and a y-intercept equal to c. The equation of the line is then:

y = mx + c

If x is the only variable, then y is a function of x:

y = f(x)

Suppose you want to plot the line. To plot a line by hand, we only need two points on it. For example, you could evaluate it at x = 1 and x = 2 (increment of 1), or x = 1 and x = 3 (increment of 2), and you would get the same line.

Mathematically, we can represent this with increments:

y = f(x) $\to$ **y = f(x +** $\boldsymbol\Delta$**x)**

y = mx + c $\to$ **y = m(x +** $\boldsymbol\Delta$**x) + c**

## Plotting Function Curves

What if you wanted to plot a more complex **curve**? Consider the function f(x) = x$^\mathsf{2}$, which represents an exponential curve.

The plan is to evaluate the function for **constant increments of x**. That gives us a bunch of points on the graph – joining them together results in a rough estimate of the curve’s shape. Mathematically, if we were to plot the curve with n points, then:

\[ \begin{aligned} p_1: {}&f(x) = x^2 \\ p_2: {}&f(x + \Delta x) = (x + \Delta x)^2 \\ p_3: {}&f(x + 2\Delta x) = (x + 2\Delta x)^2 \\ &\!\!\!\!\!\!\!\!\!\vdots \\ p_n:{}&f(x + n\Delta x) = (x + n\Delta x)^2 \end{aligned} \]

Let us try to plot the f(x) = y = sin(x) curve using this approach. Consider the interval [-90, 90] in degrees, and let us increment by 15 degrees for every point on the curve. Then, we will have n = 13 points on the curve:

\[ \begin{aligned} p_1: {}&\sin(-90^\circ) = -1 \\ p_2: {}&\sin(-75^\circ) = -0.9659 \\ p_3: {}&\sin(-60^\circ) = -0.866 \\ &\!\!\!\!\!\!\!\!\!\vdots \\ p_{11}:{}&\,\,\,\,\,\sin(60^\circ) = 0.866 \\ p_{12}:{}&\,\,\,\,\,\sin(75^\circ) = 0.9659 \\ p_{13}:{}&\,\,\,\,\,\sin(90^\circ) = 1 \end{aligned} \]

Plotting these, we get the following graph:

This approach is a chore by hand, but **graphing calculators** and other tools are quick and give us a plot in no time.

## An Example of Incrementing Function Variables

Consider the algebraic function f(x, y) = x + 2xy + y. Find the expression if both x and y are incremented by 2.

### Solution

We need to find the function value for an increment of 1 on both variables. Thus, if $\Delta$x = $\Delta$y = 2, then:

f(x, y) $\to$ f(x + $\Delta$x, y + $\Delta$y) = **f(x + 2, y + 2)**

**Substituting** **x = x + 2** and **y = y + 2** in the original expression:

f(x + 2, y + 2) = (x + 2) + 2(x + 2)(y + 2) + (y + 2)

**Expanding** the right-hand side:

f(x + 2, y + 2) = x + y + 4 + 2(xy + 2x + 2y + 4)

f(x + 2, y + 2) = x + y + 4 + 2xy + 4x + 4y + 8

**f(x + 2, y + 2) = 5x + 5y + 2xy + 12 **

That is the desired result.

*All mathematical drawings and images were created with GeoGebra.*