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“But in my opinion, all things in nature occur mathematically.” ― Rene Decartes

Intervals are an interesting part of algebra. Interval arithmetic can be used to find all the real numbers between any two numbers in a set.

Superprof’s here with a little refresher course for you. As you’ll see, intervals require a certain kind of thinking.

## What Is an Interval?

In mathematics, an interval is a set of numbers containing all the real numbers in the set.

We use the symbol R for a set of real numbers. We call the endpoints of this set a and b.

Let’s have a look at [4 ; 6]. This designates all the real numbers 4 ≤ x ≤ 6. This includes the numbers 4, 5, and 6, as all of these are greater than or equal to 4 and smaller than or equal to 6.

There are quite a few different kinds of intervals.

- Closed intervals
- Open intervals
- Half-open intervals
- Degenerate intervals
- Unbounded intervals
- Bounded or finite intervals
- Left-bounded intervals
- Right-bounded intervals

When mathematicians talk about left and right with intervals, they are referring to the minimum and maximum respectively. For example, a left-open interval has no minimum and a right-open interval has no maximum.

Let’s have a look at how mathematicians can use intervals.

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## How to Write Intervals

The limits of intervals are indicated with brackets. For open intervals and half-open intervals, square brackets are used: [a ; b] for open intervals and [a ; b and a ; b] for half-open intervals.

When writing intervals, they need to be written in ascending order. They’re never written in descending order. When the brackets are closed, it means that each of the limit points is included. When the brackets are open, it means the endpoints are not included.

Here is a set of whole numbers {0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9}. It includes all the whole numbers from 0 to 9 inclusive.

For abstract sets, you need to use letters. This is very common in algebra.

There are a few common uses:

- N designates natural whole numbers.
- Z designates relative numbers.
- D designates decimals.
- Q designates ration numbers.
- R designates real numbers.
- I designates the intersection between two sets.
- U designates the union of two sets.

You’ll also see mathematical signs to designate two sets of real numbers and how they interact.

It might seem quite complicated and abstract at first, but with practice, you’ll see that it’s quite simple and easy to use.

They can be useful for visualising mathematic data. This can be used to see how various intervals interact.

Keep trying different exercises until you get the hang of it. Intervals can save you a lot of time.

Here’s a table quickly summarising them:

Sign | Meaning |
---|---|

∈ | belongs to |

∉ | does not belong to |

∞ | is infinite |

∩ | intersection |

∪ | union |

≠ | not equal to |

≤ | less than or equal to |

≥ | greater than or equal to |

< | less than |

> | greater than |

Let’s move onto how intervals are used.

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## How to Solve an Interval

Let’s looking at solving intervals. Don’t worry! **You just need to read the interval** and plot it out to see what’s included and what’s not.

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Pay particular attention to the direction of the greater than and less than symbols. They will determine where your endpoints are.

If you’ve understood how we read and write intervals, you shouldn’t have any problems with these.

### Closed Intervals

These contain the endpoints. We can denote the different ways in which two real numbers interact with x.

[a ; b] = a ≤ x ≤ b

[a ; b] = a ≤ x < b

] a ; b] = a < x ≤ b

] a ; b [= a < x < b

When the brackets are closed, x is greater than or equal to b. When the brackets are open, x is only greater than and less than.

As you can see, there are quite a few possibilities.

### Open Intervals

When a and b are different numbers:

[a ; ∞[= x ≥ a

] a ; ∞[= x > a

] - ∞ ; b] = x ≤ b

] - ∞ ; b [= x < b

An open interval does not include the endpoints. We don’t know where the endpoints are.

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## Interactions between Intervals

The intersection between the intervals [a ; b] and [c ; d] is the set of real numbers x that’s in both [a ; b] and [c ; d]. This is denoted with ∩.

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Imagine a, b, c, and d are four positive whole numbers with an intersection I:

I=[a ; b] ∩ [c ; d] ou I=[c ; d] ∩ [a ; b]

For example:

2 ∈[0 ; 5] ∩[2 ; 6] car 2 ∈[0 ; 5] et 2 [2 ; 6]

To determine the intersection between two sets, it’s a good idea to represent them as their own set. You’ll see how to do this.

### Union of Interval Sets

This is all the real numbers in both the intervals [a ; b] and [c ; d].

The union is denoted with ∪.

This can be written as:

U=[a ; b] ∪ [c ; d] or U=[c ; d] ∪ [a ; b]

For example:

2 ∈[0 ; 5] ∪ [2 ; 6] because 2 ∈[0 ; 5]

3,8 ∈[0 ; 5] ∪ [2 ; 6] because 3,8 ∈[0 ; 6]

To determine the intersection of two interval sets, you can plot them on a number line.

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### Inequations

You need to remember that the solution to an inequation is always an interval or an empty set.

This takes the unknown x and expresses it as:

A (x) ≤ B(x) or A(x)<B(x) where x is an unknown variable.

To resolve this inequation, you need to find all the values for x that satisfied the inequality: the set of real numbers for x is the solution.

We could say that two equations are equivalent if they have the same set of solutions.

Here are the possible transformations for inequations to equivalents.

- Add or subtract the same number to or from both members.
- Multiply or divide both members by the same positive number.
- Multiply or divide both members by the same negative number.
- Expand, factorise, or reduce the members.

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### Inequalities

There are three rules for inequalities. The first is that you can always add the same number to each member of an inequality: if a≤b, then a+c≤b+c.

The second is that you can members together: if a≤b and c≤d, then a+c≤b+d.

The third rule is that you can multiply or divide each inequality by the same number.

### Absolute Value

The absolute value is the middle of the interval on a number line.

Keep in mind that the distance between a and b.

Hopefully, this is enough to get you started with intervals. As you’ll have understood, in mathematics, real numbers go from -∞ to +∞. Most of the time, we’re all interested in a set of these numbers. Interval sets are just as useful for finding the numbers that are in them as finding the numbers that are excluded.

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