Learning

If you’re required to divide a circle into 5 equal parts and are looking for a handy guide which features easy to understand, step by step instructions, simply continue reading to discover how to divide a circle into 5 equal parts.

How to divide any circle into 5 equal, uniform parts:

Step one:

Your first step should be to draw a horizontal line across your circle’s diameter. Make sure to use a ruler so that your line is 100% straight. As you don’t want to draw a wonky line, that will instantly ruin your chance of being able to accurately seperate your circle into 5 equal parts.

The easiest way to complete this step is to use a compass to draw your circle so that you’ll be able to use the center dot as a reference to draw your straight vertical line through your circle’s diameter. As with no guide, it’s quite difficult to be able to draw an even, straight horizontal line across your circle’s diameter.

Step two:

Pick up your compass again and set your compass to the radius of your circle, which should be relatively simply. Make sure to place your compass on the edge of your circle where your horizontal line, which crosses your circle’s diameter hits the right side of your circle. From this point draw two little arcs which intersect your circle’s circumference.

Step three:

Mark the top arc which you’ve just drawn as e and the bottom arc which you’ve just drawn as f. As you’ll need to refer back to these points in the future.

Step four:

Next draw a vertical line through the center point of your circle, ensuring so that your circle now features a cross. Label the point which your line touches the top of your circle as G. Starting at point G draw a diagonal line down to point H, which will be the middle point between the two arcs you drew in step three.

Step five:

Now draw a small arc on the left of your original horizontal line and it call J.

Step six:

Step your compass at point G at the top of your circle and draw a line to point J.

Step seven:

Simply draw two tiny arcs which intersect your circle’s circumfrence on the left side of your circle and the right side of your circle. Now label the arc on the left side of your circle A and the arc on the right side of your circle B.

Step eight:

Keeping your compass the same size, now move your compass to point B. As you’ll need to draw a new arc which intersects your circle’s circumference at point C.

Step nine:

It’s now time to move your compass to point C. From this point make sure to draw an arc that intersects your circle’s circumference at point d.

Step ten:

Thankfully, you’ve now reached your last instruction. All that’s left to do is to erase your construction lines and you’ll have a circle which is evenly divided into 5 distinct, even parts.

To discover the number of irrational numbers between 1 and 6 as well as bonus information about irrational numbers, simply continue reading.

How many irrational numbers are between 1 and 6?

You may be surprised to learn that there are infinite irrational numbers between 1 and 6. A surprising fact as most individuals assume that there would be a finite, definitive amount of irrational numbers between two numbers which are seemingly close together.

Why are there countless irrational numbers between 1 and 6? Remember what irrational numbers are. Irrational numbers are numbers which can’t be expressed with whole fractions which feature two whole numbers a denominator and a numerator. Common examples of irrational numbers include numbers which have infinite numbers after their decimal point.

A great example of an irrational number, which features infinite unique numbers after its decimal point is pi. Pi’s first 10 digits are 3.1415926535. As you can imagine, pi is just one example of the infinite irrational numbers between the whole numbers 1 and 6. Hopefully this simple example, illustrates how its possible that there are infinite irrational numbers between two seemingly small numbers.

So remember that while there are infinite irrational numbers between two numbers that have a larger gap such as 1 and 1,000,000, there are also infinite irrational numbers between numbers that are closer together such as 1 and 6 and 1 and 4.

Interesting facts about irrational numbers:

1. Numbers with decimal points can in fact be rational numbers

While numbers with decimal points are often irrational numbers, such as pi, be careful when it comes to defining numbers which feature decimal points as numbers with decimal points can be rational numbers. As an example, the number9.5. Remember that rational numbers can be expressed as whole fractions. Taking the above example 9.5 can be expressed as a simple fraction, 19/2. Both the denominator and the numerator in this fraction are whole numbers, which is also a requirement of fractions which can be made from rational numbers

.Another example of a decimal number, which may at first look like an irrational number, is the number .001. To see whether .001 happens to be a rational digit, you need to turn it into a fraction which features whole numbers. Expressed as a whole fraction the digit .001 can be written as 1/1000. As you’ll notice both the numerator and the denominator are whole numbers, which confirms that the example above is a rational number, even though it has a smaller value than the number 1.2.

There are a few famous examples of irrational numbers While you already know that the number pi is an irrational number, there are also other examples of famous irrational numbers. As an example, the golden ratio is another example of a famous irrational number and the first few numbers of the golden ratio are 1.61803398874948482.

If you’ve never heard of the golden ratio, the golden ratio is a special number which is commonly used in art, geometry and architecture. Hopefully you now have a greater understanding of irrational numbers and how to identify irrational numbers as well as rational numbers.

If you’re currently educating yourself on rational numbers and irrational numbers, continue reading to discover examples of rational numbers which aren’t integers.

What is an example of a rational number that is not an integer?

Firstly, make sure to define the term integer:

To discover a rational number that isn’t an integer, it may help to define the term integer. In the study of mathematics an integer is not simply a number but a whole number, rather than a fractional number. An example of a rational number which is an integer is the number one.
However, keep in mind that integers don’t have to be positive and negative whole numbers can also be classified as integers. An example of a rational integer which is a negative number is -5.
Finding a rational number that is not an integer?
Now that you fully comprehend what an integer is, it will be easier to find a rational number that isn’t an integer. A great example to keep in mind is a decimal number, which has reoccurring, repeating decimal numbers.
As while decimal figures which have unique numbers such as pi are irrational numbers, decimal numbers which have repeating numbers such as 0.36363 are rational numbers. Which means that 0.363636 is a prime example of a rational number that is not an integer.

Further examples of rational numbers that are not integers:

Using the information listed above, the following numbers which aren’t integers are also rational numbers, 0.444444, 0.242424 and 0.5555. All of which follow the rules for rational figures and which can all be easily expressed as whole fractions.

Why are repeating decimals considered rational numbers:

If you’re wondering why repeating decimals are considered rational numbers, the answer is that repeating decimals can be described as a ratio of two integers. So any decimal which uses repetitive numbers such as 0.575757 and 0.818181 will always be considered rational numbers.

At the same time all numbers which repeat themselves continuously such as 0.5555, 0.33333 and 0.8888 are always rational numbers, which aren’t classed as integers or whole numbers too.

What is the official mathematical term for a decimal number which is infinite and never ends:

There is actually an offical mathematical term for a decimal number which is infinite and never ends. In fact, both the terms reptend and repetend can be properly used in order to describe such numbers.

Why is it important to be able to recognize rational numbers:

If you’re uncertain about why it’s important to learn how to differentiate rational figures from irrational figures, remember that once you start figuring out more complex mathematics equations such as algebraic equations, the rules that you’ll need to follow in order to get the right result, will be dependent on whether you’re dealign with a whole rational number or an irrational number.

So in order to save yourself a headache in the future, it’s great to cement the basics of mathematics such as understanding key mathematics terms and rules.

Hopefully your comprehension of rational numbers, irrational numbers and integers has increased as a result of reading the above guide.

To be able to confidently and competently perform simple and complex equations of horizontal and vertical lines, it’s important to first understand the key difference between horizontal lines and vertical lines.

Better yet you’ll also discover how to use the slope intercept equation among a few different formulas, that you may use on a regular basis.

A guide to understanding horizontal and vertical lines:

The definition of a horizontal line:A horizontal line will go straight from left to right and a common example of a horizontal line is a far horizon, in the distance. Horizontal lines are said to be horizontal to the horizon. Horizontal lines will also never cross each other and so if you come across two lines that cross each other at a given point, the line which you’re looking at won’t be a horizontal line.

If you’re wondering why it’s important to learn about horizontal lines and to use horizontal lines to perform equations, the simple answer is that vertical lines will help you accurately determine if there is a relation which is a maths function.

How does a vertical line differ to a horizontal line:

A vertical line, simply goes from top to bottom. All the points on a vertical line will share the exact same x co-ordinate. A vertical line also differs from a horizontal line as unlike a horizontal line which features a slope, a vertical line doesn’t have a slope. So the slope of a horizontal line will always be written down in formulaic equations as being zero.

Equations of horizontal lines:

One of the easiest formulas which you should start off with, is the the intercept equation which will help you find the value of a slope. The slope intercept equation is simply y=mx+ b, m=0. However, if b value = your y co-ordinate of your y intercept, you’ll need to use the equation y = b.

Remember that as your y value will always have the same value, -1, the equation which you should use for your given examples y= -1.

Next up, it’s a great idea to try using the one to one test. Which is as follows, if a horizontal line crosses a function in a graph more than once, then the horizontal line in question is not a one to one function.

Equations of vertical lines:

The very first equation which you should use which involves a vertical line is the formula ofvertical lines, which is fairly straight forward and easy to understand. In an example equation a vertical line is given the expression x = k. Next, you may be interested in learning how to conduct a vertical line test. Which is also a pretty straight forward equation to solve.

In order to solve a vertical line test, start off by solving the equation y = f (x). Once you’re able to solve the equations listed above, you should have no trouble taking on more complex equations that feature horizontal lines and vertical lines. So if you’re ready to get to work start of with solving the slope intercept equation shown above!

If you’re interested in understanding more complex statistics lessons and geometry lessons, it’s a great idea to teach yourself about the use of empirical formulas and the use of endpoint formulas. If so, simply continue reading to discover everything which you need to know about both formulas.

Understanding empirical formulas and endpoint formula:

What is an empirical formula:

In the study of statistics the term empirical formula refers to a simple, mathematical formula which shows the relationship between the mean, the mode and the median. All of which are common statistical terms.

What does the term mean refer to in statistics?

The term mean is statistics refers to the average number of a group of numbers. To find the mean of a group of numbers add up all the numbers in your specified range and divide the answer which you get by the number of items which you added up.

What does the term median refer to in statistics?

The term median refers also refers to an average number. However if you were to look at a group of numbers and used the formula for the mean and the formula for the median to find your average number, you’d get two very different results.

To find the median of a set of numbers, sort your numbers from smallest to largest, then work your way inwards from the bottom number and the top number in your ordered list, until you successfully find the middle number. If you find that there are two numbers in the middle of your ordered list, your answer will be the average of the two numbers which you’re looking at.

What does the term mode refer to in statistics?

The mode in statistics, refers to the number in a list or group which appears most frequently. To find the mode of a group of numbers, it’s a great idea to use a tally chart to chart the number of times which each of your numbers appears in your list.

Why is the empirical formula important?

If you’re looking for a true average, it’s a great idea to use an empirical formula, to figure out a number which is influenced by your mode, median and mean, not just a single calculation.

What is an endpoint formula?

The endpoint formula is used by mathematicians, particularly in geometry and algebra, in order to find the endpoint of a line. In order to figure out the end point of a line, you’ll need to know the midpoint and the opposite end of your line. After which point, you’ll need to use a two line algebra equation to find your endpoint.

If you’d like to have a go at working out the end point of a line, simply use the endpoint formula which is (x2, y2) = (2_m1 – x1, 2_m2 – y1). If you’re a bit overwhelmed by the end point formula just remember to use the acronym BEDMAS, in order to work out your equation.

Hopefully you’re now confident about your understanding of the endpoint formula and the empirical formula both of which are commonly used in mathematics.

If you’ve been reading up on irrational numbers to try and understand the differences between irrational numbers and rational numbers, simply continue reading to discover a handy guide to both irrational numbers and rational numbers.

What is a rational number?

A rational number is any number which can be written down as a whole fraction. If you’re wondering what a whole fraction is, a whole fraction features two full numbers and does not feature any numbers which have decimal points.

Remember that fractions feature two numbers, a numerator and a denominator, when a rational number is described as being rational both numbers, the numerator and the denominator must be full numbers.

What is an irrational number?

Unlike a rational number, irrational numbers can’t be described using whole fractions. Many numbers which feature decimal points are irrational numbers. An obvious example being pi, which is 3.14159265359 and which can’t be expressed as a fraction with whole numbers.

However, if you assumed that all decimal figures are classed as being irrational, think again. There are examples, where numbers which feature decimal points can actually be expressed as whole fractions. So be sure to think twice before you assume that a number is an irrational number.

Can irrational numbers be negative?

If you were curious about whether or not an irrational number could be negative, the answer is yes, negative numbers can be negative or positive, just like any other number. Furthermore most negative versions of normal irrational numbers also happen to be irrational.

Is the number zero a rational or irrational number?

The number zero is a rational number as zero can be divided by itself to give a full number, zero. So if a friend tries to tell you that zero is an irrational number, make sure to teach them that zero is in fact a rational figure.

Are there any other methods that can help you find rational numbers?

Yes, if you notice then a number has recurring digits after its decimal point, the number which you’re looking at is in fact a rational number.

Rational numbers are also the numbers which you’re far more likely to use on a daily basis, such as full numbers like 1,2,3,4,5,6,7,8,9 and 10. As all full numbers happen to be rational numbers. A fact that is useful to try to remember.

Why is it necessary to know about both rational numbers and irrational numbers?

Even if you are now clear about what constitutes a rational number and what constitutes an irrational number, you may be wondering what the point of being able to tell the two apart is.

The simple answer is that to perform some complicated mathematical calculations, you’ll need to use different methods and formulas, depending on whether the number that you’re dealing with is a rational number or an irrational number.

So if you see yourself dealing with algebraic formulas in the future, it’s definitely essential to be able to identify both types of numbers. So that you’ll be able to use the right formula or equation.

Whether maths is part of your degree, your job or your business or you’re simply passionate about mathematical equations, continue reading to discover a few maths tips, that you may find useful.

5 Maths tips for budding mathematicians:

1. Create a dictionary of terms

As mathematics primarily studies numbers, you may be surprised that mathematics also leans heavily on English as when you study complicated branches of mathematics you’ll start to come across major terms that may be difficult to work out.

Every time you come across a brand new definition, it’s a great idea to write down your new term and its definition in your very own maths dictionary. You can even use an app on your phone to create a simple maths dictionary.

2. Use blank number line 10 intervals in order to show your thinking process when you work through complicated mathematical equations

The term blank number line 10 intervals, refers to a special type of mathematical paper which mathematics students use to show their thought processes when they’re completing a particularly challenging maths equations.

It’s highly worthwhile printing out your own or ordering your own test pieces of paper, to experiment with. As you may find that you find it useful being able to look back at your thought processes and how you successfully managed to solve complex problems.

Even if you get some of the problems which you work on wrong, by looking at the steps which you took and the equation which you should’ve used to get a correct answer you’ll be able to see where you went wrong and to amend your answer.

Sometimes the best way to learn as a mathematician is to learn from your mistakes. As once you make a mistake once, you’re very unlikely to make the same exact mistake a second time.

3. Use real life problems to study new mathematical concepts

Instead of dealing with purely theoretical mathematical concepts, it’s a great idea to use real life problems to study new mathematical concepts and branches. As an example, you may want to use problems which deal with political elections when you study new statistics methods. Or you might want to think about the Earth revealing around the sun, when you’re dealing with a new algebraic equation.

Most individuals find it far easier picking up new maths concepts, when they can relate the information which they learn to real life examples.

4. Use acronyms to recall the order of mathematical equations

As an example, chances are high that when you were an elementary school aged kid that your maths teacher taught you the acronym BEDMAS, which is meant to help you memorize algebra’s order of operations.

5. Make sure that you study in a quiet, relaxing environment

You’ll find it far more difficult to pick up and grasp new methods if your environment is distracting or uncomfortable. So it’s well worth ensuring that you work at your maths problems in a quiet, private, comfortable environment.

So there you have it, 5 simple tips which should help any budding mathematician learn and retain new methods without any stress.

If you’re interested in learning about velocity and some of its basic terms, simply continue reading to discover a simple guide to understanding velocity.

The average velocity definition:

If you’re keen to learn about velocity and how it works, a great first step to take is to make sure that you have a clear understanding of the term average velocity.

The term average velocity simply refers to the average or middle speed of an object which is calculated by the distance which an object has traveled by the amount of time which has gone past. The term average velocity can also be accurately defined as displacement divided by the time which has gone past.

What are some key examples of velocity which you may seek to measure:

If you’re curious about some key examples of velocity which you may want to measure, some examples include the velocity of a car on a high way and the velocity of a commercial plane, flying through the skies.

What is the main difference between the terms average velocity and velocity?

You may be wondering what the key difference is between the terms average velocity and velocity. If so, a straightforward answer is that the term velocity refers to the slope of the tangent to the distance vs time curve while the term average velocity refers to the same factors but averaged out over a preset period of time.

How to accurately calculate the average velocity:

In order to calculate the average velocity, your first step should be to write down a figure which represents the total displacement. If you’re confused by the term total displacement, just figure out the distance and direction between your two given points, your starting point and your end point.

Once you’ve jotted down a number that you’re confident with, your next step will be to write down the total amount of clasped time. Once you’ve done this, make sure that you’ve converted your answer to seconds as seconds are the international scientific standard, which should always be used. Especially if you plan on calculating the average velocity of an item for academic research.

Is there another method which can be used to calculate average velocity?

Yes, you can also try to calculate average velocity from constant acceleration. If you’re interested in trying out this particular formula, start off by making note of your initial velocity and constant acceleration. When you refer to speed make sure to use meters per second squared, to make things easier for yourself.

Now all that you have to do is to use the simple scientific formula (vf + vi)/2, to find the answer to the problem which you’re seeking to solve.

The three main types of velocity:

If you’re curious about understanding velocity further, it’s a great idea to ensure that you understand the three main types of velocity. They are as follows, constant velocity, constant acceleration and variable acceleration.

Hopefully you now have a well rounded understanding of the definition of the commonly used scientific term average velocity.

To discover how to convert any degree to radians, you’ll need to use the following formula. Start off by multiplying your value in degrees by the conversion factor which always equates to 0.017453292519943.

If you want to double check that you fully understand the formula which you’ll need to use, simply walk through the following example of how to covert 72 degrees to radians. To ensure that you come to the same accurate conclusion. Take the number 72, which refers to your given number and multiply 72 by your conversion factor which is again, 0.17453292519943.

The answer which you should get to this simple and easy to perform calculation is 1.2566370614359 radians. If your calculation is off, its time and trouble shoot. The biggest mistake which you can make when it comes to converting degrees to radians is to get a number or two out of sequence in your conversion factor. Which sometimes happens as the conversion factor does feature 13 decimal numbers. Which is a lot to remember.

As the conversion factor is so long, you may want to use an online calculator to figure out your conversion as you’ll be able to copy and paste the correct conversion factor directly from this article into your online calculator. Which will ensure that you will get the right answer to your conversions, every time. Without having to trouble shoot an incorrect answer.

You can use the same formula which was described above in order to convert any given number of degrees into radians. As an example, to convert 90 degrees into radians, all you’ll need to do is to multiply your number, which in this case is 90 by the same conversion factor, which you used above. Once you’ve got the knack of using this formula, you’ll be able to use it, time and time again with success.

Extra information which is helpful when you’re dealing with degrees and radians:

What is the definition of a degree in mathematics?

While most individuals refer to degrees in mathematics from elementary school, you may not have a clear, simple to understand definition of a degree. A single degree in the study of mathematics is simply 1 out of 360 revolutions. Therefore it should come as no surprise that a single degree measures pi divided by 180 radians.

What is a clear concise definition of the term radian?

In the study of mathematics the term radian is used to describe an angle which is subtended by a circular arc, divided by the angle of the very same arc. If you’re referring to one radian, it is an angle which is subtended by the middle of your circle and which boasts an arc which is equal in length to the radius of the circle which you’re dealing with.

A recap on how to easily transform degrees into radians:

As a quick refresher and to cement the information which you’ve just learned to convert any number of degrees into radians you must multiply your number by the conversion factor. The latter of which will always stay the same and is well worth noting down.

If you’re having a bit of difficulty successfully multiplying two different binomials, continue reading to discover a handy little definition, that you’ll find easy to remember.

What are binomials?

If you’re a bit confused by the mathematics term binomials, the term refers to a polynomial, which features not one but two terms. An example of a basic binomial is 3 x squared + 2.

What is the foil method definition:

The foil method is actually a handy acronym, that you can use in order to recall the correct order for multiplying different binomials. As in order to get accurate results, you’ll need to perform each step in the correct order. If you remember using the acronym BEDMAS in order to solve algebraic equations in high school, you’ll love the foil acronym, which works the same way.

What does each letter in the foil method definition stand for?

The letter f in the foil method stands for first, which means that you should start of multiplying your binomials by multiplying your first terms.

Next up, the letter o, in the method refers to outer and reminds you to multiply your outer terms, as your second move.

Your third letter, letter i refers to the word inner and reminds you to multiply your inner terms, as your third move in your equation.

Lastly the last letter in the world foil is l and stands for last. Lastly, make sure to multiply your last terms, in order to complete your equation.

An example to experiment with:

If you want to try out using the foil defintion use the easy example of (a+b) (c+d). In this example, your first terms are a and c. Your outer terms are a and d, your inner terms are b and c and lastly your last terms are b and d.

So if you have an example of terms and you’re not sure about how to get started, it’s well worth writing down your example’s terms and matching them up with (a+b) (c+d) equation above and using the foil definition to complete your equations.

Extra tips on multiplying polynomials:

When you’re multiplying polynomials, you’ll be required to multiply each of your terms in one polynomial by the other terms in your second polynomial. If you feel overwhelmed, don’t worry as you can always make your life a little simpler by adding together your answers in order to simplify your equations.

If you’re looking to multiply a single polynomial by another single single polynomial, simply multiply your equation’s constants by its variables, in order to get the answer to your equation.

In such equations its common to use a . instead of the symbol x, too multiply and x looks too much like the multiplication symbol.

In conclusion:

So next time that you’re tasked with multiplying two binomials, if you forget which terms to multiply first, just remember the simple acronym FOIL and you’ll soon be reminded of the correct order of terms which you should multiply. As you’ll be sure to find the right answer and to be able to answer your equation successfully and without any hassles.