Learning

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.

Whether you’re interested in studying statistics, mathematics in general or plan to undertake academic research in the near future, it’s well worth ensuring that you fully understand how and when to use empirical probability.

A simple guide to empirical probability:

Firstly, what is empirical probability?

If you’re looking for a simple, easy to understand definition you’re in luck. The term empirical probability is used in statistics as well as in scientific and academic research and refers to a branch of probability which is designed to assist individuals in being able to predict the likelihood of test results, based on data collected from previous test results on the same subject matter.

What are some of the key differences between empirical probability and theoretical probability?

If you’re unsure of some of the differences between the two common types of probabilities, you’ve come to the right place, in order to find the answers that you’re looking for. Theoretical probability compares the number of positive outcomes to the number of possible outcomes in a test. Whereas as mentioned above the study of empirical probability deals with using past data to predict the probability of favorable test results.

So as you can see, the two types of probabilities both seek to discover the likelihood of positive scenarios occurring but each type of probably takes a different approach to seeking an answer. If in doubt it’s always a wise idea to use multiple different probability calculations to see whether there is any correlation between the answers which you’ll end up with.

How to calculate empirical probability accurately:

Thankfully calculating empirical probability is simple and straightforward and involves following a straight forward empirical probability formula.

Firstly add up all the instances in which the specific events which you’re focused on have occurred and write down your answer. Next write down the number of the trials or observations which took place. Lastly, simply divide your first number by your second number. In order to find your desired answer.

Once you’ve used this simple formula once, you should be able to remember it easy, in order to use it again in the future. Especially as this particular equation only involves finding out two pieces of information and then diving one piece of information by the second piece of information.

Are there any further tips on accurately predicting empirical probability?

Yes, keep in mind that the answer which you should get should always be between zero and one. If you discover an answer of 0, this means that the event in question never occurred. While if you get an answer of 1, it’s an indication that the event in question always occurred.

Are there any other types of probability which you can use instead of empirical probability and theoretical probability?

Yes, another type of probability which is often used in the study of statistics is experimental probability.

So if you’re looking to discover the probability of a specific outcome or event, it’s well worth using the formula for empirical probability, in order to discover your likely outcome.

If you’re currently unsure of how to accurately go about calculating the empirical formula for any given compound, simply continue reading to discover a simply guide to follow. So that you’ll easily be able to calculate the empirical formula for any compound such as our example compound, which is c8h18.

What is an empirical formula:

The term empirical formula is used to express a single compound as its whole number of individual atoms, for each element which is contained in a compound. It’s important to note that each compound will boast a different empirical formula to its molecular formula. So you shouldn’t expect the two formulas to be identical or to give the same exact result.

If you’re confused, remember that a compound’s molecular formula refers to the whole number of atoms in a single molecule. Which should be relatively easy to remember.

The reason why the term empirical formula has the name empirical in it, is that every empirical formula has been determined by using scientific data which has been conducted on each scientific compound.

The empirical formula for c8h18:

If you’re looking to discover the empirical formula for any compound such as c8h18, you must follow the instructions below. In order to find an accurate answer. Better yet if you continue reading the entire article listed below, you’ll also discover c8h18’s connection with the chemical compound octane.

Step one:

Start off with listing the number of grams of each individual element which is found in the compound which you’re investigating.

Step two:

Next, in turn convert the mass of each individual element in your compound to moles. To do this you may need to study molar mass, which you’ll conveniently find on any periodic table.

Step three:

Once you’ve successfully completed step 2, you’ll need to divide each of your mole values by the smallest number of moles which you have calculated.

Step four:

Your last job is to round the number which you get to the nearest whole number. Your answer is the empirical formula for your given compound such as the example compound c8h18.

Once you’re familiar with this particular formula, you’ll be able to work out the empirical formula of any scientific compound.

What is the empirical formula for the octane:

You may be interested to read that the empirical formula for octane is c8h18. Which is the example of a compound which has been used throughout this article.

What is octane?

If you’re unfamiliar with octane, the term simply refers to a relatively common chemical compound which refers to a hydrocarbon which is paired with alkane and features a highly complex structural formula. Which is normally simplified to c8h18.

You may have come across the term octane at a gas station where each different fuel that you’ll come across will have a different octane number such as unleaded 98, 95 or 91.

After reading the article above, you should now have no issues calculating the empirical formula for any compound, that you’re interested in studying further. As you can always refer back to this easy to understand guide for reference.

To demystify the process of diving monomials, simply continue reading to discover a handy guide to follow, in order to ensure that you obtain the right answers.

How to divide monomials:

In order to be able to accurately divide monomials, your first step will be to decide whether you want to divide a monomial by another monomial or to divide a monomial by a polynomial. As both processes have different instructions to follow in order to get an accurate answer.

How to divide a monomial by another monomial:

Your first step should be to divide the coefficients of both of your monomials. Once you’ve done this, your second step entails diving your variables. If some of your exponents share the same base, you have to follow the law of exponents which specifies that you should divide by subtracting your exponents.

When you try and divide some monomials by other monomials you may need to simplify your equation in order to be able to solve it.

How to divide a monomial by a polynomial:

If you’re interested in learning how to divide a monomial to a polynomial, you’ll need to use a different, more complicated equation to find your desired answer. You’ll need to distribute a divisor that will be used to divide into either a sum or a difference.

Alternatively in some instances you’ll be able to use the number 2 to distribute into your equation and to simplify your complex equation. Once you’ve distributed the number 2 into each part of the equation which you’re working with, you’ll be able to divide your terms by the number 2.

To make things a little clearer when you divide a single monomial by a polynomial, start off by distributing the term 2x over your polynomial. You can do this by dividing each of your terms by 2x. Lastly, you should then divide each of your terms that you’re dealing with which will be a monomial with another monomial.

Extra tips on dividing monomials by polynomials:

1. Make sure that your final answers are expressed without the use of any negative exponents. If you have negative exponents in one of your final answers, you’ll need to relook at your answer and your processes in order to make sure that you’re able to come up with a final answer which doesn’t use any negative exponents.

Is there a quicker way to divide monomials by monomials and to divide monomials by polynomials?

While it’s a great idea to be able to manually work out how to divide monomials by other monomials and polynomials, so that you ensure that you understand the correct process, there is a shorter way of dividing monomials by monomials or polynomials.

The benefits of using a dividing monomials calculator:

Whether you’re still confused by the two equations which you’ll need to follow in order to be able to successfully divide monomials or you’re simply short on time, you can turn to using a dividing monomials calculator. Which will give you an accurate answer in a few seconds.

Better yet, the answers which you’ll get will never feature any negative exponents, which is a problem which you may face if you try to find your answers manually.

If you’re currently studying the key equations which are used in the study of implicit differentiation, you’ve come to the right place. Simply continue reading a guide to understanding implicit differentiation as well as some of the most commonly used terms that you’ll come across in your study of implicit differentiation. An example of which is the term second differentiation.

What does the term implicit differentiation refer to?

The term implicit differentiation refers to a chain of rules which are used to work with derivatives. If you plan on studying or brushing up on your calculus skills, you’ll find that the bulk of exercises which you’ll work through feature y functions which are written as functions of x.

What is the d 2y dx 2 implicit differentiation?

Now that you’re aware of what implicit differentiation refers to, it’s time to learn about one of its key terms. Firstly, it’s important to understand what the equation above refers to. The equation d 2y dx 2 refers to what is referred to in mathematics as the second differentiation. If you’ve never heard of second differentiation, simply continue reading to find out more valuable information.

What exactly is the second differentiation and how is it used in mathematics?

The second differentiation is a method which you can use to figure out the nature of multiple of stationary points. Key examples of stationary points which you can figure out the nature of using implicit differentiation include maximum points, minimum points and points of inflection.

What are maximum points?

If you’re unsure about what the term maximum points refers to a circumstance when the gradient almost reaches its maximum point. In this scenario, just before the maximum point the gradient is positive and which is reaches its maximum it reaches zero. After the maximum point, the same gradient will be classed as being negative.

The term maximum point refers to the middle position where the gradient exactly reaches its maximum value.

What are minimum points?

Now that you fully understand what the term maximum points refers to, it’s time to learn about their counterparts which are called minimum points. Minimum points are positive and occur when a gradient has not met its maximum and has not reset to zero.

Remember that any gradient which reaches the maximum point or which surpasses the maximum point and become negative, is not an example of a minimum point.

What is an inflection point?

In order to better understand the process of using the second differentiation equation when dealing with implicit differentiation it’s also a wise idea to understand the term inflection point. What is an inflection point? An inflection point is any point which you can locate on a graph where a tangent successfully crosses a curve.

What does the term point of reflection refer to in mathematics?

The term point of reflection refers to a case where the derivative has reached a local maximum.

Why you should use a d 2y dx 2 calculator:

The quickest way to accurately calculate second differentiation problems is to use a calculator.

Hopefully you now have a clear understanding of implicit differentiation and its use in calculus.