Engineering calculator online

We hasten to present to everyone a free engineering calculator. With it, any student can quickly and, most importantly, easily perform various kinds 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 the widest range of Internet users. Now, when you need a calculator, visit our website and use the free engineering calculator.

An engineering calculator can perform both simple arithmetic operations and rather 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, in search of simple solutions, types a query in search engines: an online mathematical calculator. The free web application will help you instantly calculate the result of any 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, PI constant. Parentheses should be used for complex calculations.

Features of the engineering calculator:

1. basic arithmetic operations;
2. work with numbers in a standard form;
3. calculation of trigonometric roots, functions, logarithms, exponentiation;
4. statistical calculations: addition, arithmetic mean or standard deviation;
5. application of a memory cell and user 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:

Extraction of roots (square root, cubic root, as well as the root of the n-th degree);
ex (e to x power), exponent;
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: base two binary logarithm is log2x, base ten base ten logarithm is log, natural logarithm is ln.

This engineering calculator also includes a calculator of quantities with the ability to convert physical quantities for various measurement systems - computer units, distance, weight, time, etc. With 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 will be useful to put the cursor in the input field). Among other things, data can be entered using the buttons of the calculator itself.

To build graphs in the input field, write the function as indicated in the example field or use the toolbar specially designed for this (to go to it, click on the button with the icon in the form of a graph). To convert values, press Unit, to work with matrices - Matrix.

First level

Expression conversion. Detailed Theory (2019)

Often we hear this unpleasant phrase: "simplify the expression." Usually, in this case, we have some kind of monster like this:

“Yes, much easier,” we say, but such an answer usually does not 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 a (just!) ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to deal with fractions and factorize polynomials.

Therefore, if you have not done this before, be sure to master the topics "" and "".

Read? If yes, then you are ready.

Let's go! (Let's go!)

Important note!If instead of formulas you see gibberish, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac)

Basic Expression Simplification Operations

Now we will analyze the main techniques that are used to simplify expressions.

The simplest of them is

1. Bringing similar

What are similar? You went through this in 7th grade, when letters first appeared in math instead of numbers.

Similar are terms (monomials) with the same letter part.

For example, in the sum, like terms are and.

Remembered?

Bring similar- means to add several similar terms with each other and get one term.

But how can we put letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects.

For example, the letter is a chair. Then what is the expression?

Two chairs plus three chairs, how much will it be? That's right, chairs: .

Now try this expression:

In order not to get confused, let different letters denote different objects.

For example, - this is (as usual) a chair, and - this is a table.

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients.

For example, in the monomial the coefficient is equal. And he is equal.

So, the rule for bringing similar:

Examples:

Bring similar:

Answers:

2. (and are 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 is needed factorize, i.e. represent as a product.

Especially this important in fractions: because in order to reduce the fraction, the numerator and denominator must be expressed as a product.

You went through the detailed methods of factoring expressions in the topic "", so here you just have to remember what you have learned.

To do this, solve a few examples (you need to factorize)

Examples:

Solutions:

3. Fraction reduction.

Well, what could be nicer than to cross out part of the numerator and denominator, and throw them out of your life?

That's the beauty of abbreviation.

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 a 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 deleted.

Examples:

The principle, I think, is clear?

I would like to draw your attention to one typical mistake in abbreviation. Although this topic is simple, but many people do everything wrong, not realizing that cut- it means divide numerator and denominator by the same number.

No abbreviations if the numerator or denominator is the sum.

For example: you need to simplify.

Some 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, so you can reduce.

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

Here is another example: .

This expression is decomposed into factors, which means that you can reduce, that is, divide the numerator and denominator by, and then by:

You can immediately divide by:

To avoid such mistakes, remember an easy way to determine if an expression is factored:

The arithmetic operation that is performed last when calculating the value of the expression is the "main".

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 decomposed into factors).

If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).

To fix it yourself, a few examples:

Examples:

Solutions:

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

The first step should be to factorize:

4. Addition and subtraction of fractions. Bringing fractions to a common denominator.

Adding and subtracting ordinary fractions is a well-known 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 coprime, 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. Here, first of all, we turn mixed fractions into improper ones, and then - according to the usual scheme:

It is quite another matter if the fractions contain letters, for example:

Let's start simple:

a) Denominators do not contain letters

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

now in the numerator you can bring similar ones, if any, and factor them:

Try it yourself:

Answers:

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 once;

and multiply them by all other factors, not common ones.

To determine the common factors of the denominators, we first decompose them into simple factors:

We emphasize the common factors:

Now we write out the common factors once and add to them all 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:

We decompose the denominators into factors;

determine common (identical) multipliers;

write out all the common factors once;

We multiply them by all other factors, not common ones.

So, in order:

1) decompose the denominators into factors:

2) determine the common (identical) factors:

3) write out all the common factors once and multiply them by all the other (not underlined) factors:

So the common denominator is 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 the extent

to the extent

to the extent

in degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere is it said 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 has been learned?

So, another unshakable rule:

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

But what do you need to multiply to get?

Here on and multiply. And multiply by:

Expressions that cannot be factorized will be called "elementary factors".

For example, is an elementary factor. - too. But - no: it is decomposed into factors.

What about 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 do the same with them.

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

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

Another example:

Decision:

Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:

Fine! Then:

Another example:

Decision:

As usual, we 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 already so similar ... And the truth is:

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 we bring to a common denominator:

Got it? Now let's check.

Tasks for independent solution:

Answers:

Here we must 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 doubled product. The incomplete square of the sum is one of the factors in the expansion of the difference of cubes:

What if there are already three fractions?

Yes, the same! First of all, we will make sure that the maximum number of factors in the denominators is the same:

Pay attention: 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 is reversed again. As a result, he (the sign in front of the fraction) has not changed.

We write out the first denominator in full in the common denominator, and then we 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 goes like this:

Hmm ... With fractions, it’s clear what to do. But what about the two?

It's simple: you know how to add fractions, right? So, you need to make sure that the deuce becomes a fraction! Remember: a fraction is a division operation (the numerator is divided by the denominator, in case you suddenly 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 now over. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numeric expression? Remember, considering the value of such an expression:

Did you count?

It should work.

So, I 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, you can do them in any order.

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

But: the parenthesized expression is evaluated out of order!

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

What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the order of actions 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 that's not the same as an expression with letters, is it?

No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations 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 it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.

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

For example:

Let's simplify the expression.

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

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

2) We get:

Multiplication of fractions: what could be easier.

3) Now you can shorten:

That's it. Nothing complicated, right?

Another example:

Simplify the expression.

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

Decision:

First of all, let's define the procedure.

First, let's add the fractions in brackets, instead of two fractions, one will turn out.

Then we will do the division of fractions. Well, we add the result with the last fraction.

I will schematically number the steps:

Now I will show the whole process, tinting the current action with red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.

2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

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

And promised at the very beginning:

Answers:

Solutions (brief):

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

Now on to learning!

EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA

Basic simplification operations:

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

  IMPORTANT: only multipliers can be reduced!

• Addition and subtraction of fractions:
  ;
• Multiplication and division of fractions:
  ;

Remark 1

A logical function can be written using a logical expression, and then you can go to the logical circuit. It is necessary to simplify logical expressions in order to get as simple as possible (and therefore cheaper) logical circuit. In fact, a logical function, a logical expression, and a logical circuit are three different languages ​​that talk about the same entity.

To simplify logical expressions, use laws of the algebra of logic.

Some transformations are similar to the transformations of formulas in classical algebra (bracketing the common factor, using commutative and associative laws, etc.), while other transformations are based on properties that classical algebra operations do not have (using the distribution law for conjunction, laws of absorption, gluing, de Morgan's rules, etc.).

The laws of the algebra of logic are formulated for basic logical operations - “NOT” - inversion (negation), “AND” - conjunction (logical multiplication) and “OR” - disjunction (logical addition).

The law of double negation means that the "NOT" operation is reversible: if you apply it twice, then in the end the logical value will not change.

The law of the excluded middle states that any logical expression is either true or false (“there is no third”). Therefore, if $A=1$, then $\bar(A)=0$ (and vice versa), which means that the conjunction of these quantities is always equal to zero, and the disjunction is equal to one.

$((A + B) → C) \cdot (B → C \cdot D) \cdot C.$

Let's simplify this formula:

Figure 3

This implies that $A = 0$, $B = 1$, $C = 1$, $D = 1$.

Answer: students $B$, $C$ and $D$ are playing chess, but student $A$ is not playing.

When simplifying logical expressions, you can perform the following sequence of actions:

1. Replace all “non-basic” operations (equivalence, implication, XOR, etc.) with their expressions through the basic operations of inversion, conjunction, and disjunction.
2. Expand inversions of complex expressions according to de Morgan's rules in such a way that only individual variables have negation operations.
3. Then simplify the expression using parentheses expansion, bracketing common factors, and other laws of the algebra of logic.

Example 2

Here, de Morgan's rule, the distributive law, the law of the excluded middle, the commutative law, the law of repetition, the again commutative law, and the law of absorption are used in succession.

With the help of 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 will talk about simplifying expressions in this lesson.

People communicate in different languages. For us, an important comparison is the pair "Russian language - mathematical language". The same information can be reported in different languages. But, besides this, it can be pronounced differently in one language.

For example: “Peter is friends with Vasya”, “Vasya is friends with Petya”, “Peter and Vasya are friends”. Said differently, but one and the same. By any of these phrases, we would understand what is at stake.

Let's look at this phrase: "The boy Petya and the boy Vasya are friends." We understand what is at stake. However, we don't like how this phrase sounds. Can't we simplify it, say the same, 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 easier, but not to lose, not to distort the meaning.

The same thing happens in mathematical language. The same thing can be said 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 multitude, we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider a 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 all the work and get the equivalent expression as a single number.

Consider an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, you must perform all the actions that are possible.

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

Example: Subtract the number from the 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: .

Decision

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

2) Calculate the products: .

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

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

To determine the equivalent expression, you must:

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: the sum does not change from the rearrangement of the terms.

2. Associative 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 the sum from a number, you can subtract each term individually.

Properties of multiplication and division

1. The commutative property of multiplication: the product does not change from a permutation of factors.

2. Associative 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. The distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Decision

1) Imagine how

2) Let's represent the first multiplier as the sum of 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 the opposite direction: .

Follow these steps:

1) 2)

Decision

1) For convenience, you can use the distribution law, just 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 in the kitchen and hallway. Kitchen area - hallway -. There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the condition of the problem

Decision

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

§ 1 The concept of simplifying a literal expression

In this lesson, we will get acquainted with the concept of “similar terms” and, using examples, we will learn how to carry out the reduction of similar terms, thus simplifying literal expressions.

Let's find out the meaning of the concept of "simplification". The word "simplification" is derived from the word "simplify". To simplify means to make simple, simpler. Therefore, to simplify a literal expression is to make it shorter, with a minimum number of actions.

Consider the expression 9x + 4x. This is a literal expression that is a sum. The terms here are presented as products of a number and a letter. The numerical factor of such terms is called the coefficient. In this expression, the coefficients will be the numbers 9 and 4. Please note that the multiplier represented by the letter is the same in both terms of this sum.

Recall the distributive law of multiplication:

To multiply the sum by a number, you can multiply each term by this number and add the resulting products.

In general, it is written as follows: (a + b) ∙ c \u003d ac + bc.

This law is valid in both directions ac + bc = (a + b) ∙ c

Let's apply it to our literal expression: the sum of the products of 9x and 4x is equal to the product, the first factor of which is the sum of 9 and 4, the second factor is x.

9 + 4 = 13 makes 13x.

9x + 4x = (9 + 4)x = 13x.

Instead of three actions in the expression, one action remained - multiplication. So, we have made our literal expression simpler, i.e. simplified it.

§ 2 Reduction of like terms

The terms 9x and 4x differ only in their coefficients - such terms are called similar. The letter part of similar terms is the same. Similar terms also include numbers and equal terms.

For example, in the expression 9a + 12 - 15, the numbers 12 and -15 will be similar terms, and in the sum of the products of 12 and 6a, the numbers 14 and the products of 12 and 6a (12 ∙ 6a + 14 + 12 ∙ 6a), the equal terms represented by the product of 12 and 6a.

It is important to note that terms with equal coefficients and different literal factors are not similar, although it is sometimes useful to apply the distributive law of multiplication to them, for example, the sum of the products of 5x and 5y is equal to the product of the number 5 and the sum of x and y

5x + 5y = 5(x + y).

Let's simplify the expression -9a + 15a - 4 + 10.

In this case, the terms -9a and 15a are similar terms, since they differ only in their coefficients. They have the same letter multiplier, and the terms -4 and 10 are also similar, since they are numbers. We add like terms:

9a + 15a - 4 + 10

9a + 15a = 6a;

We get: 6a + 6.

Simplifying the expression, we found the sums of like terms, in mathematics this is called the reduction of like terms.

If bringing such terms is difficult, you can come up with words for them and add objects.

For example, consider the expression:

For each letter we take our own object: b-apple, c-pear, then it will turn out: 2 apples minus 5 pears plus 8 pears.

Can we subtract pears from apples? Of course not. But we can add 8 pears to minus 5 pears.

We give like terms -5 pears + 8 pears. Like terms have the same literal part, therefore, when reducing like terms, it is enough to add the coefficients and add the literal part to the result:

(-5 + 8) pears - you get 3 pears.

Returning to our literal expression, we have -5s + 8s = 3s. Thus, after reducing similar terms, we obtain the expression 2b + 3c.

So, in this lesson, you got acquainted with the concept of “similar terms” and learned how to simplify literal expressions by bringing like terms.

List of used literature:

1. Mathematics. Grade 6: lesson plans for the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. Mnemosyne 2009.
2. Mathematics. Grade 6: a textbook for students of educational institutions. I.I. Zubareva, A.G. Mordkovich.- M.: Mnemozina, 2013.
3. Mathematics. Grade 6: textbook for educational institutions / G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others / edited by G.V. Dorofeeva, I.F. Sharygin; Russian Academy of Sciences, Russian Academy of Education. M.: "Enlightenment", 2010.
4. Mathematics. Grade 6: textbook for general educational institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. – M.: Mnemozina, 2013.
5. Mathematics. Grade 6: textbook / G.K. Muravin, O.V. Ant. – M.: Bustard, 2014.

Used images: