Mathematical-Calculator-Online v.1.0

The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percentage calculations and other operations.

Solution:

How to use a math calculator

Key	Designation	Explanation
5	numbers 0-9	Arabic numerals. Entering natural integers, zero. To get a negative integer, you must press the +/- key
.	semicolon)	Separator to indicate a decimal fraction. If there is no number before the point (comma), the calculator will automatically substitute a zero before the point. For example: .5 - 0.5 will be written
+	plus sign	Adding numbers (integers, decimals)
-	minus sign	Subtracting numbers (integers, decimals)
÷	division sign	Dividing numbers (integers, decimals)
X	multiplication sign	Multiplying numbers (integers, decimals)
√	root	Extracting the root of a number. When you press the “root” button again, the root of the result is calculated. For example: root of 16 = 4; root of 4 = 2
x 2	squaring	Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16
1/x	fraction	Output in decimal fractions. The numerator is 1, the denominator is the entered number
%	percent	Getting a percentage of a number. To work, you need to enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button
(	open parenthesis	An open parenthesis to specify the calculation priority. A closed parenthesis is required. Example: (2+3)*2=10
)	closed parenthesis	A closed parenthesis to specify the calculation priority. An open parenthesis is required
±	plus minus	Reverses sign
=	equals	Displays the result of the solution. Also above the calculator, in the “Solution” field, intermediate calculations and the result are displayed.
←	deleting a character	Removes the last character
WITH	reset	Reset button. Completely resets the calculator to position "0"

Algorithm of the online calculator using examples

Addition.

Addition of integers natural numbers { 5 + 7 = 12 }

Addition of whole natural and negative numbers { 5 + (-2) = 3 }

Adding decimal fractions (0.3 + 5.2 = 5.5)

Subtraction.

Subtracting natural integers ( 7 - 5 = 2 )

Subtracting natural and negative integers ( 5 - (-2) = 7 )

Subtracting decimal fractions (6.5 - 1.2 = 4.3)

Multiplication.

Product of natural integers (3 * 7 = 21)

Product of natural and negative integers ( 5 * (-3) = -15 )

Product of decimal fractions ( 0.5 * 0.6 = 0.3 )

Division.

Division of natural integers (27 / 3 = 9)

Division of natural and negative integers (15 / (-3) = -5)

Division of decimal fractions (6.2 / 2 = 3.1)

Extracting the root of a number.

Extracting the root of an integer ( root(9) = 3)

Extracting the root from decimals( root(2.5) = 1.58 )

Extracting the root of a sum of numbers ( root(56 + 25) = 9)

Extracting the root of the difference between numbers (root (32 – 7) = 5)

Squaring a number.

Squaring an integer ( (3) 2 = 9 )

Squaring decimals ((2,2)2 = 4.84)

Conversion to decimal fractions.

Calculating percentages of a number

Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )

Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )

18% of the number 140 is (140 * 0.18 = 25.2)

The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times, and since then their use has only increased. A polynomial is an algebraic sum of the products of numbers, variables and their powers. Converting polynomials typically involves two types of problems. The expression needs to be either simplified or factorized, i.e. represent it as the product of two or more polynomials or a monomial and a polynomial.

To simplify the polynomial, give similar terms. Example. Simplify the expression \ Find monomials with the same letter part. Fold them up. Write down the resulting expression: \ You have simplified the polynomial.

For problems that require factoring a polynomial, determine the common factor of the given expression. To do this, first remove from brackets those variables that are included in all members of the expression. Moreover, these variables should have the lowest indicator. Then calculate the largest common divisor each of the coefficients of the polynomial. The modulus of the resulting number will be the coefficient of the common multiplier.

Example. Factor the polynomial \ Take it out of brackets \ because the variable m is included in each term of this expression and its smallest exponent is two. Calculate the common multiplier factor. It is equal to five. Thus, the common factor of this expression is \ Hence: \

Where can I solve a polynomial equation online?

You can solve the equation on our website https://site. The free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group http://vk.com/pocketteacher. Join our group, we are always happy to help you.

First level

Converting Expressions. Detailed Theory (2019)

Converting Expressions

We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:

“It’s much simpler,” we say, but such an answer usually doesn’t work.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.

Have you read it? If yes, then you are now ready.

Basic simplification operations

Now let's look at the basic techniques that are used to simplify expressions.

The simplest one is

1. Bringing similar

What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in the sum, similar terms are and.

Do you remember?

To bring similar means to add several similar terms to each other and get one term.

How can we put the letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .

Now try this expression: .

To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.

So, the rule for bringing similar ones is:

Examples:

Give similar ones:

Answers:

2. (and similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.

You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):

Solutions:

3. Reducing a fraction.

Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?

That's the beauty of downsizing.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).

To reduce a fraction you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be crossed out.

The principle, I think, is clear?

I would like to draw your attention to one thing typical mistake when contracting. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.

No abbreviations if the numerator or denominator is a sum.

For example: we need to simplify.

Some people do this: which is absolutely wrong.

Another example: reduce.

The “smartest” will do this: .

Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.

Here's another example: .

This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:

You can immediately divide it into:

To avoid such mistakes, remember easy way how to determine whether an expression is factorized:

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To consolidate, solve a few yourself examples:

Answers:

1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be factorization:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. First thing here mixed fractions we turn them into incorrect ones, and then follow the usual pattern:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Everything here is the same as with ordinary numerical fractions: find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all common factors once and multiply them by all other (unemphasized) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent solution:

Answers:

Here we need to remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:

What to do if there are already three fractions?

Yes, the same thing! First of all, let's make sure that maximum amount the factors in the denominators were the same:

Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.

We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm... It’s clear what to do with fractions. But what about the two?

It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent the expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

OK it's all over Now. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:

Now I’ll show you the process, tinting the current action in red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Solutions (brief):

If you have coped with at least the first three examples, then you have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
• Factorization: putting the common factor out of brackets, applying it, etc.
• Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factorize
  2) if the numerator and denominator have common factors, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Adding and subtracting fractions:
  ;
• Multiplying and dividing fractions:
  ;

Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.

For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.

Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we are talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.

In mathematical language, roughly the same thing happens. One and the same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to do everything and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

1) Perform the actions in the first and second brackets: .

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

2. Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property multiplication: to multiply a number by a sum, you need to multiply it by each addend separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distributive law can also be used in reverse side: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, only use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each cost? three types linoleum? (Fig. 1)

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

Engineering calculator online

We are happy to present everyone with a free engineering calculator. With its help, any student can quickly and, most importantly, easily perform various types of mathematical calculations online.

The calculator is taken from the site - web 2.0 scientific calculator

A simple and easy-to-use engineering calculator with an unobtrusive and intuitive interface will truly be useful to a wide range of Internet users. Now, whenever you need a calculator, go to our website and use the free engineering calculator.

An engineering calculator can perform as simple arithmetic operations, and quite complex mathematical calculations.

Web20calc is an engineering calculator that has a huge number of functions, for example, how to calculate all elementary functions. The calculator also supports trigonometric functions, matrices, logarithms and even plotting.

Undoubtedly, Web20calc will be of interest to that group of people who are looking for simple solutions types in search engines the query: mathematical online calculator. A free web application will help you instantly calculate the result of some mathematical expression, for example, subtract, add, divide, extract the root, raise to a power, etc.

In the expression, you can use the operations of exponentiation, addition, subtraction, multiplication, division, percentage, and the PI constant. For complex calculations, parentheses should be included.

Features of the engineering calculator:

1. basic arithmetic operations;
2. working with numbers in a standard form;
3. calculation of trigonometric roots, functions, logarithms, exponentiation;
4. statistical calculations: addition, arithmetic mean or standard deviation;
5. use of memory cells and custom functions of 2 variables;
6. work with angles in radian and degree measures.

The engineering calculator allows the use of a variety of mathematical functions:

Extracting roots (square, cubic, and nth root);
ex (e to the x power), exponential;
trigonometric functions: sine - sin, cosine - cos, tangent - tan;
inverse trigonometric functions: arcsine - sin-1, arccosine - cos-1, arctangent - tan-1;
hyperbolic functions: sine - sinh, cosine - cosh, tangent - tanh;
logarithms: binary logarithm to base two - log2x, decimal logarithm to base ten - log, natural logarithm - ln.

This engineering calculator also includes a quantity calculator with the ability to convert physical quantities for various systems measurements - computer units, distance, weight, time, etc. Using this function, you can instantly convert miles to kilometers, pounds to kilograms, seconds to hours, etc.

To make mathematical calculations, first enter a sequence of mathematical expressions in the appropriate field, then click on the equal sign and see the result. You can enter values ​​directly from the keyboard (for this, the calculator area must be active, therefore, it would be useful to place the cursor in the input field). Among other things, data can be entered using the buttons of the calculator itself.

To build graphs, you should write the function in the input field as indicated in the field with examples or use the toolbar specially designed for this (to go to it, click on the button with the graph icon). To convert values, click Unit; to work with matrices, click Matrix.

Return