What do you associate with the number 3.14? Correct! It is the number pi (\(\pi\)) or, strictly speaking, only the first digits of \(\pi\). In ancient times, the Greek mathematician Archimedes succeeded in calculating \(\pi\) with an approximation method that gave him exactly these two decimal places. You will learn how he got to the number \(\pi\), what the relation to geometry is and how people use multiples of pi in this article.
- What is the Number pi?
- How did Archimedes calculate the Number pi?
- Whar are multiples of pi?
- Odd multiples of pi
- Even multiples of pi
- Multiples of pi in decimals
- Multiples of pi/4
- Examples of Multiples of Pi
What is the Number Pi?
The number pi is an irrational number represented by the Greek letter (\(\pi\), pronounced /paɪ/). An irrational number is just a number with an infinite number of decimal places. Pi represents the ratio of the circumference of a circle to its diameter.
Fig. 1. Diagram showing the ratio of the circumference of a circle to its diameter
Because \(\pi\) is an irrational number, it is impossible to know every digit of \(\pi\). However, mathematicians have calculated about 82 trillion decimal digits of this number. It is usually approximated to 3.14 or 3.1416.
The number pi is also called Archimides' constant because he was the one to note the relevance of \(\pi\) in geometry and calculate its approximation in the 3rd century BC.
How did Archimedes Calculate the Number Pi?
The Greek mathematician Archimedes is considered to be the discoverer of the number \(\pi\). In the year 250 B.C., he was able to calculate up to two decimal places of \(\pi\) through approximation. How did he manage to do this?
In a circle, \(\pi\) corresponds exactly to half the circumference (C) of the circle.
\(\pi = \frac{C}{d}\)
Archimedes drew a unit circle, i.e. a circle with a radius of \(r = 1\) or a diameter of \(d = 2\). Inside and outside of this Unit Circle, Archimedes drew a regular hexagon, whose perimeter he could calculate.
Fig. 2. Approximation method to calculate \(\pi\) with Unit Circle and hexagons.
The circumference C of the circle is thus larger than the perimeter of the inner hexagon, but smaller than that of the outer hexagon.
Archimedes was able to apply the relation we mentioned earlier, that in the unit circle \(\pi = \frac {C}{2}\), adding to that the known perimeters of both hexagons, so that he could determine a lower and upper limit for the number \( \pi\).
\(\frac{P_{inner hexagon}}{2} < \pi < \frac{P_{outer hexagon}}{2}\)
\(3 < \pi < 3.464101615\)
P is the symbol for the perimeter.
To get a better approximation of the number \(\pi\), Archimedes divided the sides of the original hexagons to make a dodecagon (12 sides), then an icositetragon (24 sides) and so forth until he reached an enneacontahexagon (96 sides).
In this way, he could determine the following limits of \(\pi\):
\(3.1408450 < \pi < 3.1428571\)
What are Multiples of Pi?
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 (a whole number, i.e. a number with no decimals). 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.
Pi is a number, therefore it can be multiplied by other numbers, including Integers. Thus, finding the multiples of pi is done the same way as finding the multiples of any number.
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.
Odd Multiples of Pi
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 \( 1 \), \( 3 \), \( 5 \), \( 7 \), \( 9 \), etc.
From the definition above, we see that to find the odd multiples of \( \pi \), you will have to multiply \( \pi \) 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.
Even Multiples of Pi
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.
Multiples of Pi in Decimals
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
\[ \begin{array}{lll} \pi \cdot 1 & \approx 3.142 \cdot 1 &= 3.142 , \\ \pi \cdot 2 & \approx 3.142 \cdot 2 &= 6.284, \\ \pi \cdot 3 & \approx 3.142 \cdot 3 & = 9.426 .\end{array} \]
Multiples of Pi/4
The significance of this very specific number, Pi/4, often written \( \frac{\pi}{4}\), comes from the use of \(\pi\) in trigonometry.Remember that \(\pi\) is always the result of dividing the circumference C of a circle by its diameter d (\(\pi = \frac{C}{d}\)). Therefore, \(C = d\pi\). In a unit circle, d = 2: \(C = 2\pi\). Therefore, \(C = d\pi\). In a unit circle, d = 2: \(C = 2\pi\).
When measuring the degree of an angle using \(\pi\), we talk of Radians instead of degrees. A full circle is 360º, and also \(2\pi\). Therefore, \(360^\circ \equiv 2\pi\).
Thus, \(180^\circ \equiv \pi, \quad 90^\circ \equiv \frac{\pi}{2} \quad and \quad 45^\circ \equiv \frac{\pi}{4}\).
Pi/4 is a fraction and therefore it is possible to find its multiples like you would 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:
\(\begin{align} \frac{\pi}{4} \cdot 1 &= \frac{\pi}{4} \equiv 45^\circ, \\ \frac{\pi}{4} \cdot 2 &= \frac{\pi}{2} \equiv 90^\circ, \\ \frac{\pi}{4} \cdot 3 &= \frac{3\pi}{4} \equiv 135^\circ, \\\frac{\pi}{4} \cdot 4 &= \pi \equiv 180^\circ,\\ \frac{\pi}{4} \cdot 5 &= \frac{5\pi}{4} \equiv 225^\circ. \end{align} \)
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.
Examples of Multiples of Pi
Let's look at some examples of 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 of 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 - Key takeaways
- 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 \( 3.14159 \).
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