Degree of number: definitions, designation, examples. Raising a number to a negative power

Obviously, numbers with powers can be added like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2 .
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4 .

Odds the same powers of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2 .

It is also obvious that if we take two squares a, or three squares a, or five squares a.

But degrees various variables and various degrees identical variables, must be added by adding them to their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3 .

It is obvious that the square of a, and the cube of a, is neither twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6 .

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahend must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Power multiplication

Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3 .

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n is;

And a m , is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are - negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y-n .y-m = y-n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y).(a + y) = a 2 - y 2 .
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4 .
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8 .

Division of powers

Numbers with powers can be divided like other numbers by subtracting from the divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is a 3 .

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing powers with the same base their scores are subtracted..

So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master the multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Reduce the exponents in $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.

The exponent is used to make it easier to write the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation of such a transition is given in the first section of this article). Powers make it easier to write long or complex expressions or equations; also, powers are easily added and subtracted, resulting in a simplification of an expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to decide exponential equation(in such an equation the unknown is in the exponent), read .

Steps

Solving simple problems with powers

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a problem with exponents manually, rewrite the exponent as a multiplication operation, where the base of the exponent is multiplied by itself. For example, given the degree 3 4 (\displaystyle 3^(4)). In this case, the base of degree 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two quadruples, and then replace them with the result. Like this:

• 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
  • 4 ∗ 4 = 16 (\displaystyle 4*4=16)

1. Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:

   • 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
     • 16 ∗ 4 = 64 (\displaystyle 16*4=64)
   • 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
     • 64 ∗ 4 = 256 (\displaystyle 64*4=256)
   • 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
     • 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
   • Keep multiplying the result of multiplying the first two numbers by the next number until you get the final answer. To do this, multiply the first two numbers, and then multiply the result by the next number in the sequence. This method is valid for any degree. In our example, you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .

2. Solve the following problems. Check your answer with a calculator.

   • 8 2 (\displaystyle 8^(2))
   • 3 4 (\displaystyle 3^(4))
   • 10 7 (\displaystyle 10^(7))

3. On the calculator, look for the key labeled "exp", or " x n (\displaystyle x^(n))", or "^". With this key you will raise a number to a power. It is practically impossible to manually calculate the degree with a large exponent (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; to do this, click "View" -\u003e "Engineering". To switch to normal mode, click "View" -\u003e "Normal".

   • Check your answer with search engine(Google or Yandex). Using the "^" key on the computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for study).

   Addition, subtraction, multiplication of powers

   1. You can add and subtract powers only if they have the same base. If you need to add powers with the same bases and exponents, then you can replace the addition operation with a multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented as 1 ∗ 4 5 (\displaystyle 1*4^(5)); thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply such a degree and this number. In our example, raise 4 to the fifth power, and then multiply the result by 2. Remember that the addition operation can be replaced by a multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:

      • 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
      • 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
      • 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
      • 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))

   2. When multiplying powers with the same base, their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. In this way, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:

      When raising a power to a power, the exponents are multiplied. For example, given a degree. Since the exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The meaning of this rule is that you multiply the power (x 2) (\displaystyle (x^(2))) on itself five times. Like this:

      • (x 2) 5 (\displaystyle (x^(2))^(5))
      • (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
      • Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))

   3. An exponent with a negative exponent should be converted to a fraction (to the inverse power). It doesn't matter if you don't know what a reciprocal is. If you are given a degree with a negative exponent, for example, 3 − 2 (\displaystyle 3^(-2)), write this power in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:

      When dividing powers with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). In this way, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

      • The degree in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.

   4. Below are some expressions to help you learn how to solve power problems. The above expressions cover the material presented in this section. To see the answer, just highlight the empty space after the equals sign.

   Solving problems with fractional exponents

   A degree with a fractional exponent (for example, ) is converted to a root extraction operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x(\displaystyle(\sqrt(x))). It does not matter what number is in the denominator of the fractional exponent. For example, x 1 4 (\displaystyle x^(\frac (1)(4))) is the fourth root of "x" x 4 (\displaystyle (\sqrt[(4)](x))) .

   1. If the exponent is an improper fraction, then such an exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule for multiplying powers. For example, given a degree. Turn that exponent into a root whose exponent is equal to the denominator of the fractional exponent, and then raise that root to the exponent equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:

      • x 5 3 (\displaystyle x^(\frac (5)(3)))
      • x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
      • x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
   2. Some calculators have a button for calculating exponents (first you need to enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
   3. Remember that any number is equal to itself to the first power, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, for example, 5 ∗ 1 = 5 (\displaystyle 5*1=5) and 5 / 1 = 5 (\displaystyle 5/1=5).
   4. Know that the degree 0 0 does not exist (such a degree has no solution). When you try to solve such a degree on a calculator or on a computer, you will get an error. But remember that any number to the power of zero is equal to 1, for example, 4 0 = 1. (\displaystyle 4^(0)=1.)
   5. In higher mathematics, which operates with imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax), where i = (− 1) (\displaystyle i=(\sqrt (())-1)); e is a constant approximately equal to 2.7; a is an arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.

   Warnings

   • As the exponent increases, its value greatly increases. Therefore, if the answer seems wrong to you, in fact it may turn out to be true. You can check this by plotting any exponential function, such as 2 x .

As you know, in mathematics there are not only positive numbers, but also negative ones. If acquaintance with positive degrees begins with determining the area of ​​​​a square, then with negative ones everything is somewhat more complicated.

This should be known:

1. Raising a number to a natural power is the multiplication of a number (the concept of a number and a figure in the article will be considered equivalent) by itself in such an amount as the exponent (in what follows we will use the word indicator in parallel and simply). 6^3 = 6*6*6 = 36*6 =216. AT general view it looks like this: m^n = m*m*m*…*m (n times).
2. It should be borne in mind that when a negative number is raised to a natural power, it will become positive if the exponent is even.
3. Raising a number to an exponent of 0 gives a unit, provided that it is not equal to zero. Zero to the power of zero is considered undefined. 17^0 = 1.
4. Extracting a root of a certain degree from a number is called finding a number that, when raised to an appropriate indicator, will give the desired value. So the cube root of 125 is 5 because 5^3 = 125.
5. If you want to raise a number to a positive fractional power, then you need to raise the number to the denominator and extract the root of the numerator from it. 6^5/7 = 7th root of 6*6*6*6*6.
6. If you want to raise a number to a negative exponent, then you need to find the reciprocal of this one. x^-3 = 1/x^3. 8^-4 = 1/8^4 = 1/8*8*8*8 = 1/4096.

Raising a number to a negative power modulo from zero to one

First, we must remember what is a module. This is the distance on the coordinate line from the value we have chosen to the origin (zero of the coordinate line). By definition, it can never be negative.

Value greater than zero

With a value of a digit in the range from zero to one, a negative indicator gives an increase in the digit itself. This happens because the denominator decreases, while remaining positive.

Let's look at examples:

• 1/7^-3 = 1/(1/7^3) = 1/(1/343) = 343;
• 0,2^-5 = 1/0,2^5 = 1/0,2*0,2*0,2*0,2*0,2 = 1/0,00032 = 3125.

Moreover, the larger the module of the indicator, the more actively the figure grows. As the denominator tends to zero, the fraction itself tends to plus infinity.

Value less than zero

Now let's look at how to raise to a negative power if the number is less than zero. The principle is the same as in the previous part, but the sign of the exponent matters here.

Let's look at the examples again:

• -19 / 21^-4 = 1/(-19/21)^4 = 1/(-19)^4/21^4 = 21^4/(-19)^4 = 21*21*21*21/(-19)*(-19)*(-19)*(-19) = 194481/130321 = 1,4923228;
• -29/40^-5 = 1/(-29/40)^5 = 1/(-29)^5/40^5 = 40^5/(-29)^5 = 40*40*40*40*40/(-29)*(-29)*(-29)*(-29)*(-29) = 102400000/(-20511149) = -4,9924.

AT this case, we see that module continues to grow, but the sign depends on whether the exponent is even or odd.

It should be noted that if we build a unit, it will always remain itself. If you need to raise a number minus one, then with an even exponent it will turn into one, with an odd one it will remain minus one.

Raising to a negative integer power if the modulus is greater than one

For digits whose modulus more than one, have their own characteristics of action. First of all, you need to convert the whole part of the fraction to the numerator, that is, convert it to an improper fraction. If we have decimal, then it must be converted to normal. This is done as follows:

• 6 integers 7/17 = 109/17;
• 2,54 = 254/100.

Now consider how to raise a number to a negative power under these conditions. Already from the above, we can assume what we should expect from the result of the calculations. Since the double fraction is reversed during simplifications, the modulus of the digit will decrease the faster, the greater the modulus of the indicator.

First, consider the situation where the given number is positive.

First of all, it becomes clear that the final result will be greater than zero, because dividing two positives always gives a positive one. Again, let's look at examples of how this is done:

• 6 integer 1/20 to the minus fifth power = 121/20^-5 = 1/(121/20)^5 = 1/121^5/20^5 = 20^5/121^5 = 3200000/25937424601 = 0 .0001234;
• 2,25^-6 = (225/100)^-6 = 1/(225/100)^6 = 1/225^6/100^6 = 100^6/225^6 = 100*100*100*100*100*100/225*225*225*225*225*225 = 0,007413.

As we see, special difficulties actions do not cause, and all our initial assumptions turned out to be true.

Now we turn to the case of a negative digit.

To begin with, we can assume that if the indicator is even, then the result will be positive, if the indicator is odd, then the result will be negative. All our previous calculations in this part will be considered valid now. Let's take a look at the examples again:

• -3 integer 1/2 to the minus sixth power = (-7/2)^-6 = 1/(-7/2)^6 = 1/(-7)^6/2^6 = 2*2*2 *2*2*2/(-7)*(-7)*(-7)*(-7)*(-7)*(-7) = 64/117649 = 0.000544;
• -1,25^-5 = (-125/100)^-5 = 1/(-125/100)^5 = 1/(-125)^5/100^5 = 100^5/(-125)^5 = 100*100*100*100*100/(-125)*(-125)*(-125)*(-125)*(-125) = 10000000000/(-30517578125) = -0.32768.

Thus, all our reasoning turned out to be correct.

Raising in the case of a negative fractional exponent

Here you need to remember that such an erection exists extracting the root of the degree of the denominator from the number in the degree of the numerator. All our previous reasoning remains true this time as well. Let's explain our actions with an example:

• 4^-3/2 = 1/4^3/2 = 1/rad(4^3) = 1/rad64 = 1/8.

In this case, you need to keep in mind that extracting the roots high level it is possible only in a specially selected form and, most likely, you will not be able to get rid of the sign of the radical (square root, cubic root, and so on) with accurate calculations.

Nevertheless, having studied the previous chapters in detail, one should not expect difficulties in school calculations.

It should be noted that the description of this chapter also includes erection with a deliberately irrational exponent, for example, if the indicator is minus PI. You need to act according to the principles described above. However, calculations in such cases become so complex that only powerful electronic computers can do it.

Conclusion

The action we studied is one of the most the most difficult tasks in mathematics(especially in the case of a fractionally rational or irrational value). However, having studied this instruction in detail and step by step, you can learn how to do it fully automatically without any problems.

In the fifth century BC ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, to come to a common opinion about the essence of paradoxes scientific community has not yet succeeded ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia logical paradox it is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What do I want to focus on Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for exploration.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.

2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic characters to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that's not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.

As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of ​​a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of numbers. Because we can't compare numbers with different units measurements. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then it has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

Then it is not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

One of the main characteristics in algebra, and indeed in all mathematics, is a degree. Of course, in the 21st century, all calculations can be carried out on an online calculator, but it is better to learn how to do it yourself for the development of brains.

In this article, we will consider the most important issues regarding this definition. Namely, we will understand what it is in general and what are its main functions, what properties exist in mathematics.

Let's look at examples of what the calculation looks like, what are the basic formulas. We will analyze the main types of quantities and how they differ from other functions.

Let's understand how to solve using this quantity various tasks. We will show with examples how to raise to a zero degree, irrational, negative, etc.

Online exponentiation calculator

What is the degree of a number

What is meant by the expression "raise a number to a power"?

The degree n of a number a is the product of factors of magnitude a n times in a row.

Mathematically it looks like this:

a n = a * a * a * …a n .

For example:

• 2 3 = 2 in the third step. = 2 * 2 * 2 = 8;
• 4 2 = 4 in step. two = 4 * 4 = 16;
• 5 4 = 5 in step. four = 5 * 5 * 5 * 5 = 625;
• 10 5 \u003d 10 in 5 step. = 10 * 10 * 10 * 10 * 10 = 100000;
• 10 4 \u003d 10 in 4 step. = 10 * 10 * 10 * 10 = 10000.

Below is a table of squares and cubes from 1 to 10.

Table of degrees from 1 to 10

Below are the results of the construction natural numbers to positive powers - "from 1 to 100".

Ch-lo	2nd grade	3rd grade
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	279
10	100	1000

Degree properties

What is characteristic of such a mathematical function? Let's look at the basic properties.

Scientists have established the following signs characteristic of all degrees:

• a n * a m = (a) (n+m) ;
• a n: a m = (a) (n-m) ;
• (a b) m =(a) (b*m) .

Let's check with examples:

2 3 * 2 2 = 8 * 4 = 32. On the other hand 2 5 = 2 * 2 * 2 * 2 * 2 = 32.

Similarly: 2 3: 2 2 = 8 / 4 = 2. Otherwise 2 3-2 = 2 1 =2.

(2 3) 2 = 8 2 = 64. What if it's different? 2 6 = 2 * 2 * 2 * 2 * 2 * 2 = 32 * 2 = 64.

As you can see, the rules work.

But how to be with addition and subtraction? Everything is simple. First exponentiation is performed, and only then addition and subtraction.

Let's look at examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 - 3 2 = 25 - 9 = 16

But in this case, you must first calculate the addition, since there are actions in brackets: (5 + 3) 3 = 8 3 = 512.

How to produce computing in more difficult cases ? The order is the same:

• if there are brackets, you need to start with them;
• then exponentiation;
• then perform operations of multiplication, division;
• after addition, subtraction.

There are specific properties that are not characteristic of all degrees:

1. The root of the nth degree from the number a to the degree m will be written as: a m / n .
2. When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
3. When building a work different numbers to a power, the expression will correspond to the product of these numbers to a given power. That is: (a * b) n = a n * b n .
4. When raising a number to a negative power, you need to divide 1 by a number in the same step, but with a “+” sign.
5. If the denominator of a fraction is in a negative power, then this expression will be equal to the product of the numerator and the denominator in a positive power.
6. Any number to the power of 0 = 1, and to the step. 1 = to himself.

These rules are important in individual cases, we will consider them in more detail below.

Degree with a negative exponent

What to do with a negative degree, that is, when the indicator is negative?

Based on properties 4 and 5(see point above) it turns out:

A (- n) \u003d 1 / A n, 5 (-2) \u003d 1/5 2 \u003d 1/25.

And vice versa:

1 / A (- n) \u003d A n, 1 / 2 (-3) \u003d 2 3 \u003d 8.

What if it's a fraction?

(A / B) (- n) = (B / A) n , (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.

Degree with a natural indicator

It is understood as a degree with exponents equal to integers.

Things to remember:

A 0 = 1, 1 0 = 1; 2 0 = 1; 3.15 0 = 1; (-4) 0 = 1…etc.

A 1 = A, 1 1 = 1; 2 1 = 2; 3 1 = 3…etc.

Also, if (-a) 2 n +2 , n=0, 1, 2…then the result will be with a “+” sign. If a a negative number raised to an odd power, vice versa.

General properties, and all the specific features described above, are also characteristic of them.

Fractional degree

This view can be written as a scheme: A m / n. It is read as: the root of the nth degree of the number A to the power of m.

With a fractional indicator, you can do anything: reduce, decompose into parts, raise to another degree, etc.

Degree with irrational exponent

Let α be an irrational number and А ˃ 0.

To understand the essence of the degree with such an indicator, Let's look at different possible cases:

• A \u003d 1. The result will be equal to 1. Since there is an axiom - 1 is equal to one in all powers;

А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 are rational numbers;

• 0˂А˂1.

In this case, vice versa: А r 2 ˂ А α ˂ А r 1 under the same conditions as in the second paragraph.

For example, the exponent is the number π. It is rational.

r 1 - in this case it is equal to 3;

r 2 - will be equal to 4.

Then, for A = 1, 1 π = 1.

A = 2, then 2 3 ˂ 2 π ˂ 2 4 , 8 ˂ 2 π ˂ 16.

A = 1/2, then (½) 4 ˂ (½) π ˂ (½) 3 , 1/16 ˂ (½) π ˂ 1/8.

Such degrees are characterized by all the mathematical operations and specific properties described above.

Conclusion

Let's summarize - what are these values ​​for, what are the advantages of such functions? Of course, first of all, they simplify the lives of mathematicians and programmers when solving examples, since they allow minimizing calculations, reducing algorithms, systematizing data, and much more.

Where else can this knowledge be useful? At any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.