What is the module of the number 2. Module of the number (absolute value of the number), definitions, examples, properties

The modulus of a number is the distance from this number to zero on the coordinate line.

The module is designated with the symbol: | |.

• Record |6| read as "modulus of number 6", or "module of six".
• Record |8| reads "module 8".

The modulus of a positive number is equal to the number itself. For example, |2| = 2. The modulus of a negative number is equal to the opposite number<=>|-3| = 3. The modulus of zero is equal to zero, that is, |0| = 0. The modules of opposite numbers are equal, that is, |-a| = |a|.

For a better understanding of the topic: “modulus of a number”, we suggest using the association method.

Let's imagine that the modulus of a number is a bath, and the minus sign is dirt.

Being under the module sign (that is, in the “bath”), the negative number is “washed”, and comes out without a “minus” sign - clean.

In the bath can "wash" (that is, stand under the sign of the module) and negative, and positive numbers, and the number zero. However, being “pure” positive numbers , and zero do not change their sign when leaving the “bath” (that is, from under the sign of the module)!

The history of the modulus of the number or 6 interesting facts about the modulus of the number

1. The word "module" comes from the Latin name modulus, which means the word "measure" in translation.
2. This term was introduced by the student of Isaac Newton, the English mathematician and philosopher Roger Cotes (1682 - 1716).
3. The great German physicist, inventor, mathematician and philosopher Gottfried Leibniz in his works and writings used the module function, which he designated mod x.
4. The module designation was introduced in 1841 by a German mathematician
Karl Weierstrass (1815 - 1897).
5. When writing a module, it is denoted by the symbol: | |.
6. Another version of the term "module" was introduced in 1806 by the French
a mathematician named Jean Robert Argan (1768-1822). But it is not so.
Early nineteenth century mathematician Jean Robert Argán (1768 - 1822)
and Augustin Louis Cauchy (1789 - 1857) introduced the concept of "modulus of a complex number",
which is studied in the course of higher mathematics.

Solving problems on the topic "Module of number"

Task number 1. Arrange the expressions: -|12|, 0, 54, |-(-2)|, -17 in ascending order.

— | 12 | = — 12
| — (— 2) | = 2

17 < -12 < 0 < 2 < 54, что будет равносильно:
-17 < -|12| < 0 < | — (— 2) | < 54.

Answer: -17< -|12| < 0 < | — (— 2) | < 54.

Task number 2. It is necessary to arrange the expressions: -|-14|, -|30|, |-16|, -21, | -(-9) |
in descending order.

First, let's open the brackets and modules:

— | — 14| = — 14
— |30| = -30
|-16| = 16
| -(-9) | = 9

16 > 9 > -14 > - 21 > - 30 which will be equivalent to:
|-16| > | -(-9) | > — | — 14| > — 21 > — |30|.

Answer: |-16| > | -(-9) | > - | — 14| > — 21 > — |30|

In this article, we will analyze in detail the absolute value of a number. We will give various definitions of the modulus of a number, introduce notation and give graphic illustrations. In this case, we consider various examples of finding the modulus of a number by definition. After that, we list and justify the main properties of the module. At the end of the article, we will talk about how the modulus of a complex number is determined and found.

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Modulus of number - definition, notation and examples

First we introduce modulus designation. The module of the number a will be written as , that is, to the left and to the right of the number we will put vertical lines that form the sign of the module. Let's give a couple of examples. For example, modulo -7 can be written as ; module 4,125 is written as , and module is written as .

The following definition of the module refers to, and therefore, to, and to integers, and to rational and irrational numbers, as to the constituent parts of the set of real numbers. We will talk about the modulus of a complex number in.

Definition.

Modulus of a is either the number a itself, if a is a positive number, or the number −a, the opposite of the number a, if a is a negative number, or 0, if a=0 .

The voiced definition of the modulus of a number is often written in the following form , this notation means that if a>0 , if a=0 , and if a<0 .

The record can be represented in a more compact form . This notation means that if (a is greater than or equal to 0 ), and if a<0 .

There is also a record . Here, the case when a=0 should be explained separately. In this case, we have , but −0=0 , since zero is considered a number that is opposite to itself.

Let's bring examples of finding the modulus of a number with a given definition. For example, let's find modules of numbers 15 and . Let's start with finding . Since the number 15 is positive, its modulus is, by definition, equal to this number itself, that is, . What is the modulus of a number? Since is a negative number, then its modulus is equal to the number opposite to the number, that is, the number . Thus, .

In conclusion of this paragraph, we give one conclusion, which is very convenient to apply in practice when finding the modulus of a number. From the definition of the modulus of a number it follows that the modulus of a number is equal to the number under the sign of the modulus, regardless of its sign, and from the examples discussed above, this is very clearly visible. The voiced statement explains why the modulus of a number is also called the absolute value of the number. So the modulus of a number and the absolute value of a number are one and the same.

Modulus of a number as a distance

Geometrically, the modulus of a number can be interpreted as distance. Let's bring determination of the modulus of a number in terms of distance.

Definition.

Modulus of a is the distance from the origin on the coordinate line to the point corresponding to the number a.

This definition is consistent with the definition of the modulus of a number given in the first paragraph. Let's explain this point. The distance from the origin to the point corresponding to a positive number is equal to this number. Zero corresponds to the origin, so the distance from the origin to the point with coordinate 0 is zero (no single segment and no segment that makes up any fraction of the unit segment needs to be postponed in order to get from point O to the point with coordinate 0). The distance from the origin to a point with a negative coordinate is equal to the number opposite to the coordinate of the given point, since it is equal to the distance from the origin to the point whose coordinate is the opposite number.

For example, the modulus of the number 9 is 9, since the distance from the origin to the point with coordinate 9 is nine. Let's take another example. The point with coordinate −3.25 is at a distance of 3.25 from point O, so .

The sounded definition of the modulus of a number is a special case of defining the modulus of the difference of two numbers.

Definition.

Difference modulus of two numbers a and b is equal to the distance between the points of the coordinate line with coordinates a and b .

That is, if points on the coordinate line A(a) and B(b) are given, then the distance from point A to point B is equal to the modulus of the difference between the numbers a and b. If we take point O (reference point) as point B, then we will get the definition of the modulus of the number given at the beginning of this paragraph.

Determining the modulus of a number through the arithmetic square root

Sometimes found determination of the modulus through the arithmetic square root.

For example, let's calculate the modules of the numbers −30 and based on this definition. We have . Similarly, we calculate the modulus of two-thirds: .

The definition of the modulus of a number in terms of the arithmetic square root is also consistent with the definition given in the first paragraph of this article. Let's show it. Let a be a positive number, and let −a be negative. Then and , if a=0 , then .

Module Properties

The module has a number of characteristic results - module properties. Now we will give the main and most commonly used of them. When substantiating these properties, we will rely on the definition of the modulus of a number in terms of distance.

Let's start with the most obvious module property − modulus of a number cannot be a negative number. In literal form, this property has the form for any number a . This property is very easy to justify: the modulus of a number is the distance, and the distance cannot be expressed as a negative number.

Let's move on to the next property of the module. The modulus of a number is equal to zero if and only if this number is zero. The modulus of zero is zero by definition. Zero corresponds to the origin, no other point on the coordinate line corresponds to zero, since each real number is associated with a single point on the coordinate line. For the same reason, any number other than zero corresponds to a point other than the origin. And the distance from the origin to any point other than the point O is not equal to zero, since the distance between two points is equal to zero if and only if these points coincide. The above reasoning proves that only the modulus of zero is equal to zero.

Move on. Opposite numbers have equal modules, that is, for any number a . Indeed, two points on the coordinate line, whose coordinates are opposite numbers, are at the same distance from the origin, which means that the modules of opposite numbers are equal.

The next module property is: the modulus of the product of two numbers is equal to the product of the modules of these numbers, i.e, . By definition, the modulus of the product of numbers a and b is either a b if , or −(a b) if . It follows from the rules of multiplication of real numbers that the product of moduli of numbers a and b is equal to either a b , , or −(a b) , if , which proves the considered property.

The modulus of the quotient of dividing a by b is equal to the quotient of dividing the modulus of a by the modulus of b, i.e, . Let us justify this property of the module. Since the quotient is equal to the product, then . By virtue of the previous property, we have . It remains only to use the equality , which is valid due to the definition of the modulus of the number.

The following module property is written as an inequality: , a , b and c are arbitrary real numbers. The written inequality is nothing more than triangle inequality. To make this clear, let's take the points A(a) , B(b) , C(c) on the coordinate line, and consider the degenerate triangle ABC, whose vertices lie on the same line. By definition, the modulus of the difference is equal to the length of the segment AB, - the length of the segment AC, and - the length of the segment CB. Since the length of any side of a triangle does not exceed the sum of the lengths of the other two sides, the inequality , therefore, the inequality also holds.

The inequality just proved is much more common in the form . The written inequality is usually considered as a separate property of the module with the formulation: “ The modulus of the sum of two numbers does not exceed the sum of the moduli of these numbers". But the inequality directly follows from the inequality , if we put −b instead of b in it, and take c=0 .

Complex number modulus

Let's give determination of the modulus of a complex number. Let us be given complex number, written in algebraic form , where x and y are some real numbers, representing, respectively, the real and imaginary parts of a given complex number z, and is an imaginary unit.

The absolute value of a number a is the distance from the origin to the point BUT(a).

To understand this definition, we substitute instead of a variable a any number, for example 3 and try to read it again:

The absolute value of a number 3 is the distance from the origin to the point BUT(3 ).

It becomes clear that the module is nothing more than the usual distance. Let's try to see the distance from the origin to point A( 3 )

The distance from the origin of coordinates to point A( 3 ) is equal to 3 (three units or three steps).

The modulus of a number is indicated by two vertical lines, for example:

The modulus of the number 3 is denoted as follows: |3|

The modulus of the number 4 is denoted as follows: |4|

The modulus of the number 5 is denoted as follows: |5|

We looked for the modulus of the number 3 and found out that it is equal to 3. So we write:

Reads like: "The modulus of three is three"

Now let's try to find the modulus of the number -3. Again, we return to the definition and substitute the number -3 into it. Only instead of a dot A use new point B. point A we have already used in the first example.

The modulus of the number is 3 call the distance from the origin to the point B(—3 ).

The distance from one point to another cannot be negative. Therefore, the modulus of any negative number, being a distance, will also not be negative. The module of the number -3 will be the number 3. The distance from the origin to the point B(-3) is also equal to three units:

Reads like: "The modulus of a number minus three is three"

The modulus of the number 0 is 0, since the point with coordinate 0 coincides with the origin, i.e. distance from origin to point O(0) equals zero:

"The modulus of zero is zero"

We draw conclusions:

• The modulus of a number cannot be negative;
• For a positive number and zero, the modulus is equal to the number itself, and for a negative one, to the opposite number;
• Opposite numbers have equal modules.

Opposite numbers

Numbers that differ only in signs are called opposite. For example, the numbers −2 and 2 are opposites. They differ only in signs. The number −2 has a minus sign, and 2 has a plus sign, but we don’t see it, because plus, as we said earlier, is traditionally not written.

More examples of opposite numbers:

Opposite numbers have equal modules. For example, let's find modules for −2 and 2

The figure shows that the distance from the origin to the points A(−2) and B(2) equal to two steps.

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Instruction

If the modulus is represented as a continuous function, then the value of its argument can be either positive or negative: |х| = x, x ≥ 0; |x| = - x, x

z1 + z2 = (x1 + x2) + i(y1 + y2);
z1 - z2 = (x1 - x2) + i(y1 - y2);

It is easy to see that addition and subtraction of complex numbers follow the same rule as addition and .

The product of two complex numbers is:

z1*z2 = (x1 + iy1)*(x2 + iy2) = x1*x2 + i*y1*x2 + i*x1*y2 + (i^2)*y1*y2.

Since i^2 = -1, the end result is:

(x1*x2 - y1*y2) + i(x1*y2 + x2*y1).

The operations of raising to a power and extracting a root for complex numbers are defined in the same way as for real ones. However, in the complex domain, for any number, there are exactly n numbers b such that b^n = a, that is, n roots of the nth degree.

In particular, this means that any algebraic equation of the nth degree in one variable has exactly n complex roots, some of which may be and .

Related videos

Sources:

• Lecture "Complex numbers" in 2019

The root is an icon that denotes the mathematical operation of finding such a number, the raising of which to the power indicated before the root sign should give the number indicated under this very sign. Often, to solve problems in which there are roots, it is not enough just to calculate the value. We have to carry out additional operations, one of which is the introduction of a number, variable or expression under the root sign.

Instruction

Determine the exponent of the root. An indicator is an integer indicating the power to which the result of calculating the root must be raised in order to obtain the root expression (the number from which this root is extracted). Exponent of the root, specified as a superscript before the root icon. If this one is not specified, it is a square root whose power is two. For example, the root exponent √3 is two, the exponent ³√3 is three, the root exponent ⁴√3 is four, and so on.

Raise the number that you want to add under the root sign to the power equal to the exponent of this root, which you determined in the previous step. For example, if you need to enter the number 5 under the sign of the root ⁴√3, then the exponent of the root is four and you need the result of raising 5 to the fourth power 5⁴=625. You can do this in any way convenient for you - in your mind, using a calculator or the corresponding services posted.

Enter the value obtained in the previous step under the root sign as a multiplier of the radical expression. For the example used in the previous step with adding under the root ⁴√3 5 (5*⁴√3), this action can be done like this: 5*⁴√3=⁴√(625*3).

Simplify the resulting radical expression, if possible. For the example from the previous steps, this is that you just need to multiply the numbers under the root sign: 5*⁴√3=⁴√(625*3)=⁴√1875. This completes the operation of adding a number under the root.

If there are unknown variables in the problem, then the steps described above can be done in a general way. For example, if you want to introduce an unknown variable x under the fourth degree root, and the root expression is 5/x³, then the entire sequence of actions can be written as follows: x*⁴√(5/x³)=⁴√(x⁴*5/x³)= ⁴√(x*5).

Sources:

• what is the root sign called

Real numbers are not enough to solve any quadratic equation. The simplest of quadratic equations that do not have roots among real numbers is x^2+1=0. When solving it, it turns out that x=±sqrt(-1), and according to the laws of elementary algebra, extract the root of an even degree from a negative numbers it is forbidden.