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What is an Isosceles Triangle? (Instant Answer)

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What is an Isosceles Triangle? (Instant Answer)

What is an Isosceles Triangle?

Everything You Need to Know About Isosceles Triangles

When it comes to learning about triangles, there are a handful of different types of triangles with very specific characteristics that you will need to be familiar with—and one of the most important and most common are isosceles triangles.

In this short guide, we will tell you everything you need to know about isosceles triangles, including the isosceles triangle definition, so that you can easily an isosceles triangle and solve problems involving isosceles triangles.

So, if you need an answer to the question “what is an isosceles triangle?”, then you are in the right place! You can click here to see an instant answer or you can continue reading through this short guide for a more in-depth explanation of the features and characteristics of isosceles triangles.

You can also use the quick links below to jump to a particular section of this guide:

Let’s start with a super quick review of triangles and definitions.

What is a Triangle?

By definition, a triangle is a polygon with three sides, three corners, and three interior angles. While any three-sided polygon that satisfies this definition can be called a triangle, there are several types of triangles with specific characteristics.

Examples of different types of triangles include equilateral triangles, right triangles, and, of course, isosceles triangles.

All of the different types of triangles are shown in Figure 01 for your reference.

 

Figure 01: Types of Triangles

 

What is an Isosceles Triangle?

Isosceles Triangle Definition: An Isosceles Triangle is a triangle that has two sides of equal length.

Pretty simple, right?

By definition, an isosceles triangle is a triangle that has two sides of equal length.

Any triangle that has at least two sides of equal length can be considered an isosceles triangle.

In Figure 02 below, you can see three different examples of isosceles triangles. The notches on the sides indicate that they are equal in length to each other.

 

Figure 02: The isosceles triangle definition states that a triangle is isosceles when it has two equal sides.

 

What are the Properties of Isosceles Triangles?

Now that you know the basic definition of an isosceles triangle, let’s dive deeper into the properties of isosceles triangles and their sides and angles.

Key Properties of Isosceles Triangles

  • In addition to having two equal sides, the angles opposite of those two equal sides are also equal.

  • An altitude drawn from the base of an isosceles triangle to its vertex will always be perpendicular to that base and will dive the base into two congruent segments.

These key features are illustrated in Figure 03 below.

 

Figure 03: Properties of Right Triangles

 

Figure 03 above shows isosceles triangle △EFG. Notice that sides EF and EF are congruent (which makes this triangle isosceles by definition) and that ∠EFG and ∠EGF are also congruent.

Additionally, the altitude EH is perpendicular to the base segment FG, dividing FG into two congruent halves: segments FH and GH.

These properties apply to any isosceles triangle. If you understand these properties, you can apply them to any math problem involving an isosceles triangle!

Now that you know what is an isosceles triangle and the isosceles triangle definition, you are ready to learn about a special types of isosceles triangle—namely an isosceles right triangle (i.e. an isosceles triangle with one 90-degree angle).


What is an Isosceles Right Triangle?

An isosceles right triangle is a special type of isosceles triangle where the vertex angle is a right angle (i.e. it is equal to 90 degrees), and the two congruent angles are both equal to 45 degrees.

The isosceles right triangle is often referred to as a 45-45-90 right triangle in reference to the three angle measures being 45 degrees, 45 degrees, and 90 degrees.

These key characteristics of the isosceles right triangle are shown in Figure 04 below.

 

Figure 04: The properties of an isosceles right triangle.

 

We decided to conclude this guide by featuring the isosceles right triangle since its properties are so unique and the fact that the isosceles right triangle commonly shows up on math problems, so math students should be familiar with them.

Notice in Figure 04 above that the isosceles right triangle meets the definition of an isosceles triangle since it has two equal sides (sides AB and CB are congruent) and the angles opposite those sides are also congruent (∠BAC and ∠BCA are congruent).

However, in addition to meeting the criteria for being an isosceles triangle, an isosceles right triangle has a vertex angle that is equal to 90 degrees (i.e. it has one right angle) and the two congruent angles are both equal to 45 degrees, which is why the isosceles right triangle is often referred to as a 45-45-90 right triangle.


 
 

Conclusion: What is an Isosceles Triangle?

If you came to this guide wondering “what is an isosceles triangle?”, you now know the isosceles triangle definition as well as the key characteristics and properties of isosceles triangles.

Key Takeaway: A triangle is isosceles if it has two equal sides.

Beyond this basic isosceles triangle definition, we explored the properties and relationships between the sides and angles of isosceles triangles, namely that:

  • Every isosceles triangle has at least two equal sides.

  • The angles opposite the equal sides are also congruent to each other.

  • The altitude drawn from the base of an isosceles triangle to its vertex will always be perpendicular to that base and will dive the base into two congruent segments.

There are also a few special types of isosceles triangles, especially the isosceles right triangle (also known as the 45-45-90 triangle). The isosceles right triangle is a special case where the vertex of the triangle is a 90 degree angle and the two congruent angles are both equal to 45 degrees.

In conclusion, isosceles triangles are polygons with interesting symmetrical properties that give math students incredible opportunities to explore and learn two-dimensional figures and how they relate to real life.


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Is 0 a Whole Number? (Instant Answer)

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Is 0 a Whole Number? (Instant Answer)

Is 0 a Whole Number? Yes or No?

What is the definition of a whole number and is 0 a whole number?

 

Is 0 a Whole Number? (Image: Mashup Math MJ)

 

When learning about the different categories of numbers (natural numbers, whole numbers, integers, etc.), zero can sometimes get lost in the shuffle. Many students often wonder: “Is 0 a whole number?” If you are asking this question yourself, you can click here for an instant answer or you can read through this short guide for a complete explanation, which includes definitions to some important math vocabulary terms related to numbers.

Before we determine whether or not 0 is a whole number, let’s do a quick review of the difference between natural numbers and whole number in math.

Natural Numbers

In math, a natural number is a number that can be used for counting or ordering values or amounts. The set of natural numbers starts at 1 and continues as follows: { 1, 2, 3, 4, 5, 6, 7, …}

Notice that 0 is not included in this set and, thus, 0 is not a natural number.

Whole Numbers

In math, a whole number is any number that does not include fractions, decimals, or negatives. Another way to think about the set of whole numbers is a set that includes all of the positive integers as well as zero. The set of whole numbers starts at 0 and continues as follows: { 0, 1, 2, 3, 4, 5, 6, 7, …}

Notice that 0 is indeed included in this set and, thus, we can conclude that…

 

Figure 01: The Universe of Number: Is 0 a Whole Number?

 

Is 0 a Whole Number?

Instant Answer: Zero is a Whole Number

Yes, zero is a whole number.

Why? By definition, 0 is not included in the set of natural numbers (i.e. zero is not a natural number), but it is included in the set of whole numbers (i.e. zero is a whole number).

Additionally, 0 is the first whole number and is followed by 1, 2, 3, 4, 5, etc.

The graphic in Figure 02 below shows how zero is included in the set of whole numbers but not in the set of natural numbers.

 

Figure 02: The Universe of Number: Is 0 a Whole Number?

 

Zero's Important Place in the Number System

Why is zero such an important number when it comes to the universe of numbers? We have already established that, by definition, zero is a whole number. As far as values go, zero is unique because it represents the absence of value or a null quantity (i.e. zero represents nothing, which is why it is neither positive nor negative)

So, why is 0 a whole number? Because the concept of counting whole values relies on 0 to serve as a starting point (i.e. the point where you have nothing). Without this starting point, all non-zero whole numbers would lose their context.

If this explanation is hard to understand, we can think about zero’s role as a whole number from a practical standpoint. For example, if you were measuring the number of days that you visited the gym last year, you would rely on 0 to represent the instance where you never attended at all. Then, the remaining whole numbers could be used to determine how many times you actually attended. And since you can’t visit the gym negative times or a fraction/decimal of a time, the set of whole numbers would be used for this particular scenario.

 

Is 0 a Whole Number? Yes! (Image: Mashup Math FP)

 

More Key Facts About Zero

If you were surprised to find out that 0 is a whole number, then you may also be surprised by some other interesting and fundamental facts about zero.

Here are a few examples of some surprising facts about zero,

Zero is a whole number.

Is 0 a whole number? As previously stated, zero is, by definition, a whole number, meaning that it is included in the set of whole numbers {0, 1, 2, 3, 4, 5, …}

Zero is an integer.

Is 0 an integer? Since the set of integers includes all of the whole numbers and their negative counterparts, we can also say that zero is an integer as well.

Zero is neither negative nor positive.

Is 0 positive or negative? The set of integers includes every non-zero whole number and its negative counterpart as well as zero. This definition is worded this way because zero is the only integer that is neither negative nor positive. Zero is important in this sense because it is the “neutral value” that separates all of the positive numbers from the negative numbers.

Zero is an even number.

Is 0 even or odd? While you can’t divide a number by zero, you can divide zero by a number. By definition, a number is even if it can be divided by 2 without any remainder. Since 0/2 = 0, we can say that, by definition, zero is an even number (and that zero is not an odd number).

 
 

Conclusion: Is 0 a Whole Number?

If you find yourself wondering “Is 0 a Whole Number?”, then it’s important that you understand the mathematical definition of a whole number in the first place.

In math, a whole number is any number that does not include fractions, decimals, or negatives. And, since 0 does not include fractions or decimals and is not negative, we know that 0 is included in the set of whole number {0, 1, 2, 3, 4, 5, 6, 7, …}.

In short, the answer to the question “Is 0 a Whole Number?” is yes! By definition, zero is included in the set of whole numbers and it plays the important role of the absence of value or a null quantity. All of the non-zero whole numbers have a value that is determined based on its distance from zero on the number line. Pretty cool, right?

Additionally, while zero is not a natural number, it is all of the following:

  • zero is a whole number

  • zero is an integer

  • zero is neither positive nor negative

  • zero is an even number

Whether you came to this page looking for a simple yes or no answer or a deep exploration of the properties of 0, we hope that you leave here with a greater appreciation of 0 and its status in the universe of real numbers. As you continue to learn zero’s place in the number system, you will continue to gain a deeper understanding of mathematics and numbers in general.

 
 

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Image: Mashup Math MJ

Why Am I So Bad at Math? (And How to Get Better)

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What is a Whole Number in Math?

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What is a Whole Number in Math?

What is a Whole Number in Math?

What is a whole number in math, is 0 a whole number, and how can you tell if a number is a whole number or not?

 

What is a whole number in math?

 

Do you remember when you first started learning how to count? At this early stage, you likely used your fingers as a simple counting tool. One, two, three, four, and so on. While the days of counting on your fingers are likely long behind you, the journey that you began then has led you to this point, where you are ready to learn about whole numbers, what they are, and how they fit into the number system.

Before we dive into learning about whole numbers, lets quickly review the definition of a natural number so that you can understand the difference between a natural number and a whole number later on.

What is a Natural Number?

In math, natural numbers are the numbers that we use for counting and ordering values or amounts. The set of natural numbers starts at 1 and is as follows: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …}

Natural numbers are sometimes referred to as counting numbers. Notice that the set of natural numbers does not include 0, fractions/decimals, or negative numbers.

We can visualize the natural numbers on a number line as shown in Figure 01 below:

 

Figure 01: What is a Natural Number?

 

What is a Whole Number?

Now that you know what a natural number is, you can extend that understanding to whole numbers.

In math, whole numbers are a set of numbers that includes all of the natural numbers as well as 0.

The set of whole numbers starts at 0 and is as follows: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …}

Notice that, just like the set of natural numbers, the set of whole numbers does not include fractions/decimals or negative numbers.

We can visualize the whole numbers compared to the natural numbers on a number line as shown in Figure 02 below:

 

Figure 02: What is a whole number?

 

Simply put, the set of whole numbers is just the entire set of natural numbers with zero included.

With this in mind, we can say that the set of natural numbers is a subset of the set of whole numbers, which is why the diagram in Figure 03 below is often used to demonstrate this relationship.

 

Figure 03: Natural numbers are a subset of whole numbers. All whole numbers (except zero) are natural numbers too.

 

The Role of Zero: Is 0 a Whole Number?

Next, let's explore commonly asked question regarding whole numbers: Is zero a whole number? The answer to this quest is yes!

Zero is a whole number.

When it comes to the universe of numbers (and its subsets), the value zero is important because it represents a quantity of “nothing” or an empty set. In the case of whole numbers, zero is the dividing point that separates the positive numbers from the negative numbers (since zero is neither positive or negative).

As you continue to study the universe of numbers, you will continue to consider negative numbers and the set of integers, which includes all zero, all of the natural numbers, and their negative counterpart.

The set of integers in comparison to whole numbers and natural numbers, as well as the role of zero, is shown in Figure 04 below.

The set of integers does not include any values with fractional or decimal part. It does not have a starting point and is as follows {…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…}

Figure 04: Is zero a whole number? Zero is a whole number and an integer as well.

Figure 05: Whole numbers and natural numbers are subsets of integers.

Examples of Whole Numbers

Now you are ready to extend your understanding of whole numbers to a few examples with real-world context:

  • Example #1: The number of members in the chess club. You can’t have a fraction of a person or a negative person as a member of a club, so the number used to describe the number of members will always be a whole number such as 7 or 12. And, of course, if nobody joins the chess club, you could say that there are zero members.

  • Example #2: The amount of marbles in a bag. You can’t have a fraction of a marble or a negative marble, so the number used to describe the amount of marbles in a bag will always be a whole number such as 22 or 60. And, if the bag is empty, you could say that the amount of marbles is zero.

  • Example #3: The number of cars for sale at a dealership. Again, it is not possible to have a fraction of a car or a negative car, so the number of cars for sale will be a whole number such as 5 or 116. If all of the cars are sold and there is nothing for sale at the dealership at any point in time, then the amount of cars for sale can be zero.

 

Figure 07: Whole Numbers in the Real-World: The number of cars available for sale at a dealership will always be a whole number such as 25 since it’s impossible to have a fraction of a car or a negative car and the amount of cars available can be zero if the lot is empty. Photo by Alex Suprun on Unsplash

 

Conclusion: What is a Whole Number in Math?

In math, the universe of numbers can be broken into several subsets.

The most basic of these subsets are the Natural Numbers (also known as counting numbers), which related to elementary finger counting as follows: {1, 2, 3, 4, 5, …}.

Whole numbers are a set of numbers that includes all of the natural numbers as well as 0.

Unlike the set of natural numbers, which starts at 1, the set of whole numbers starts at 0 and is as follows: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …}. Whole numbers can not be fractions or negative.

The set of whole numbers is just the entire set of natural numbers with zero included and we can say that the natural numbers are a subset of whole numbers.

While not a natural number, zero is a whole number and it plays an important role in the universe of numbers as a divider/boundary between the positive numbers and the negative numbers.

This understanding of whole numbers will help you as you continue on with your study of real numbers and their subsets, especially your next likely destination: integers.

 
 

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