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Multiples of Pi

- Calculus
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One of the best ways to make something stick to memory is repetition. This is why children are made to do additions, subtraction, multiplication and division over and over again in schools. They are the foundation of mathematics and they need to be remembered forever. Even though most people are not fans of mathematics, they are still able to do their additions and multiplications.

In this article, you will be learning about **multiples of pi**, odd and even multiples of pi, and then take some examples.

Let's, first of all, understand what is meant by multiples of a number.

The **multiple of a ****number** is the product gotten when you multiply that number by an integer. You can say it is the times' tables of that number.

Let's take a look at an example of multiples of a number.

Some of the multiples of \( 2 \) are \( 2 \), \( 4 \), \( 6 \), \( 8 \) and so on. These are gotten by multiplying \( 2 \) with positive integers.

Now let's see what pi is.

**Pi** is a mathematical constant. It is the ratio of the circumference of a circle to its diameter.

Pi is a number. An **approximated numerical value** for pi is \( 3.14159265359 \). The number is longer than that but is often approximated to \( 3.142 \) . Therefore, finding the multiples of pi is done the same way as finding the multiples of any number.

The symbol for pi is \( \pi \).

\( \pi \) has been approximated to over 31.4 trillion decimal places.

From the definition of multiples of a number and pi given above, we now know what multiples of pi mean.

**Multiples of pi** is the product gotten when you multiply pi by an integer.

Let's take a look at what happens when you multiply \( \pi \) by an odd number.

The **odd multiples** of \( \pi \) are all the multiples of pi obtained from multiplying \( \pi \) by odd numbers.

Odd numbers are numbers that aren't divisible by 2. Examples are \( 1 \), \( 3 \), \( 5 \), \( 7 \), \( 9 \) .....

From the definition above, we see that to find the odd multiples of \( \pi \), you will have to multiply by an odd number. When doing this, there may be no need to multiply with the numerical value of \( \pi \). You can just use the symbol and treat it as algebra. See the example below.

Some odd multiples of \(\pi \) are:

\[ \begin{align} \pi \cdot 1 &= \pi, \\ \pi \cdot 3 &= 3\pi , \\ \pi \cdot 5 &= 5\pi , \\ \dots \end{align} \]

and the list goes on!

You can see that the odd multiples are gotten by multiplying \(\pi \) by odd numbers.

What about multiplying \( \pi \) by an even number?

The **even multiples** of \( \pi \) are all the multiples of \( \pi \) that are obtained from multiplying \( \pi \) by even numbers.

Even numbers are numbers that are divisible by two. They can be divided into two equal pairs or parts. Examples are \( 2 \), \( 4 \), \( 6 \), \( 8 \), \( 10 \) ......

From the above definitions, we see that to find the even multiples of \( \pi \), you will have to multiply by an even number. When doing this, there may be no need to use the numerical approximation of \( \pi \). You can just use the symbol and treat it as algebra. Take a look at the example below.

Some even multiples of \(\pi \) are

\[ \begin{align} \pi \cdot 2 &= 2\pi , \\ \pi \cdot 4 &= 4\pi , \\ \pi \cdot 6 &= 6\pi ,\\ \dots \end{align}\]

You can see that the even multiples are gotten by multiplying \(\pi \) by even numbers.

Sometimes you will want an approximate value when multiplying by \( \pi \) rather than the exact one. Here you are going to use the numerical value of \( \pi \) to find its multiples. The approximate value of \( \pi \) itself is a decimal number, so, finding its multiples this way will result in a decimal number.

You can also find the odd or even multiples of \( \pi \) in decimal. All you have to do is identify the multiples that are odd or even. The example below shows how to get the multiples of \(\pi \).

Some decimal multiples of the approximation of \(\pi \) are

\[ \begin{array}{lll} \pi \cdot 1 & \approx 3.142 \cdot 1 &= 3.142 , \\ \pi \cdot 2 & \approx 3.142 \cdot 2 &= 6.284, \\ \pi \cdot 3 & \approx 3.142 \cdot 3 & = 9.426 .\end{array} \]

Pi/4, often written \( \frac{\pi}{4}\), is a fraction and it is possible to find the multiples of a fraction. All you have to do is multiply by integers.

Let's take a look at the first five multiples of \(\frac{\pi}{4} \).

The first five multiples of \( \frac{\pi}{4} \) are:

\[ \begin{align} \frac{\pi}{4} \cdot 1 &= \frac{\pi}{4} , \\ \frac{\pi}{4} \cdot 2 &= \frac{\pi}{2} , \\ \frac{\pi}{4} \cdot 3 &= \frac{3\pi}{4} , \\ \frac{\pi}{4} \cdot 5 &= \frac{5\pi}{4}. \end{align} \]

You can also find multiples of \(\frac{\pi}{2} \), \(\frac{pi}{3} \), \(\frac{\pi}{6} \) and any other fraction in terms of \( \pi \) in exactly the same way as in the above example.

Let's look at some examples on multiples of \( \pi \).

What are the first five multiples of \( \pi \)?

**Solution.**

To get the first five multiples of \( \pi \), we will multiply \( \pi \) by the integers \( 1 \), \( 2 \), \( 3 \), \( 4 \) and \( 5 \).

\[ \begin{align} \pi \cdot 1 & = \pi \\\pi \cdot 2 & = 2\pi \\\pi \cdot 3 & = 3\pi \\\pi \cdot 4 & = 4\pi \\\pi \cdot 5 & = 5\pi \\ \end{align} \]

Therefore, the first five multiples are \(\pi \), \(2\pi \), \(3\pi \), \(4\pi \) and \(5\pi \). Notice that the first five multiples of \( \pi \) include both even and odd multiples.

Let's take another example.

What are the first three odd multiples of \( \pi \)?

**Solution.**

To get the first three odd multiples of \( \pi \), you multiply \( \pi \) by \( 1 \), \( 3 \) and \( 5 \) which are odd numbers.

\[ \begin{align} \pi \cdot 1 & = \pi \\ \pi \cdot 3 & = 3\pi \\ \pi \cdot 5 & = 5\pi \\ \end{align} \]

Therefore, the first three odd multiples are \(\pi \), \(3\pi \) and \(5\pi \).

Let's look at another example.

List some of the multiples of \( \pi \) in decimals.

**Solution.**

To get some of the multiples of \( \pi \) in decimals, you will need to multiply by the numerical value of \( \pi \) which is approximately \( 3.142 \).

\[ \begin{align} \pi \cdot 1 & \approx 3.142 \\ \pi \cdot 2 & \approx 6.284 \\ \pi \cdot 3 & \approx 9.426 \\ \pi \cdot 4 & \approx 12.568 \\ \pi \cdot 5 & \approx 15.71 \\ \pi \cdot 6 & \approx 18.852 \\ \end{align} \]

Therefore, some of the multiples of \( \pi \) in decimals are: \( 3.142 \), \( 6.284 \), \( 9.426 \), \( 15.71 \) and \( 18.852 \) \(\dots \)

Let's take an example on the even multiples of \(\pi \).

List the first four even multiples of \( \frac{pi}{4} \).

**Solution.**

To find the even multiples of \( \frac{pi}{4} \), we will have to multiply by \( 2 \), \( 4 \), \( 6 \) and \( 8 \) which are all even numbers.

\[ \begin{align} \frac{\pi}{4} \cdot 2 & = \frac{\pi}{2} \\ \frac{\pi}{4} \cdot 4 & = \pi \\ \frac{\pi}{4} \cdot 6 & = \frac{3\pi}{2} \\ \frac{\pi}{4} \cdot 8 &= 2\pi \\ \end{align} \]

Therefore, the first \( 4 \) even multiples are: \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \) and \( 2\pi \).

Let's see one more example.

List some multiples of \( \frac{\pi}{2} \).

**Solution.**

Some of the multiples of \( \frac{\pi}{2} \) are:

\[ \begin{align} \frac{\pi}{2} \cdot 1 &= \frac{\pi}{2} \\ \frac{\pi}{2} \cdot 2 &= \pi \\ \frac{\pi}{2} \cdot 3 &= \frac{3\pi}{2} \\ \frac{\pi}{2} \cdot 4 &= 2\pi \\ \frac{\pi}{2} \cdot 5 &= \frac{5\pi}{2} \\ \frac{\pi}{2} \cdot 6 &= 3\pi \\ \frac{\pi}{2} \cdot 7 &= \frac{7\pi}{2} \\ \end{align} \]

Therefore, some of the multiples of \( \frac{\pi}{2} \) are: \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), \( 2\pi \), \( \frac{5\pi}{2} \), \( 3\pi \), \( \frac{7\pi}{2} \) ..... and the list goes on!

- Multiples of \( \pi \) is the product gotten when you multiply pi by an integer.
- The odd multiples of \( \pi \) are all the multiples of \( \pi \) that are obtained from multiplying \( \pi \) by odd numbers.
- When you multiply \( \pi \) with an odd number, the result is an odd multiple of \( \pi \) .
- The even multiples of \( \pi \) are all the multiples of \( \pi \) that are obtained from multiplying \( \pi \) by even numbers.
- When you multiply \( \pi \) with an even number, the result is an even multiple of \( \pi \).
- Approximations of \( \pi \) include \( \frac{22}{7}\) and \( 3.14159 \).

Multiples of pi is the product gotten when you multiply pi by an integer.

Giving a answer as a multiple of pi means to leave the answer in terms of pi. For example, if you are asked to find the area of a circle with a radius of 2 cm and give your answer as multiple of pi, you will calculate for the answer without using the numerical value of pi.

Area of a circle = pi x r^2

= pi x 2^2

= 4pi

The answer still has pi in it.

Integral multiples of pi are values gotten when pi is multiplied by integers.

More about Multiples of Pi

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