Números Complexos Nivel I

1. A provável origem do conjunto dos números complexos é mais bem associada

A insuficiência do conjunto dos números reais em responder às necessidades aritméticas da época (final do século XVII)

Ao gênio inventivo de alguns cientistas e filósofos (século XII)

à criatividade e ao ócio dos cientistas no final da idade média

Ao acaso e à inspiração de matemáticos renascentistas.

The likely origin of the set of complex numbers is better associated with the insufficiency of the set of real numbers in meeting the arithmetic needs of the time (late 17th century). This suggests that the concept of complex numbers was developed as a solution to the limitations of real numbers in certain mathematical calculations and problems.

Explanation

The likely origin of the set of complex numbers is better associated with the insufficiency of the set of real numbers in meeting the arithmetic needs of the time (late 17th century). This suggests that the concept of complex numbers was developed as a solution to the limitations of real numbers in certain mathematical calculations and problems.

2. Se o número complexo Z é tal que Z = x + yi, x real e y real, i a unidade imaginária, Im(Z) é

X

Y

Yi

I

The question is asking for the imaginary part of a complex number Z, which is represented as x + yi. The imaginary part is denoted as y, so the correct answer is y.

Explanation

The question is asking for the imaginary part of a complex number Z, which is represented as x + yi. The imaginary part is denoted as y, so the correct answer is y.

3. Hoje podemos afirmar que i é (a notação a^b equivale a "a" elevado a "b")

A unidade imaginária, e i^2 = -1

Uma letra que vale -1

Um operador que elevado ao quadrado é igual a 1

A parte imaginária de um número complexo

i is the imaginary unit, which is defined as the square root of -1. When i is squared, it results in -1. Therefore, i^2 = -1. This is a fundamental concept in complex numbers and is used in various mathematical calculations involving imaginary and complex numbers.

Explanation

i is the imaginary unit, which is defined as the square root of -1. When i is squared, it results in -1. Therefore, i^2 = -1. This is a fundamental concept in complex numbers and is used in various mathematical calculations involving imaginary and complex numbers.

4. Dentre outras coisas, a unidade imaginária trouxe significado matemático para

Os números perplexos

As raizes quadradas de números negativos

As letras numéricas como e, i e pi

Os números complexos

The correct answer is "as raizes quadradas de números negativos" which translates to "the square roots of negative numbers". The imaginary unit, represented by the symbol "i", is defined as the square root of -1. It was introduced in mathematics to handle the square roots of negative numbers, which were previously considered "imaginary" or non-existent. The imaginary unit allows for the manipulation and calculation of complex numbers, which have both a real and imaginary component.

Explanation

The correct answer is "as raizes quadradas de números negativos" which translates to "the square roots of negative numbers". The imaginary unit, represented by the symbol "i", is defined as the square root of -1. It was introduced in mathematics to handle the square roots of negative numbers, which were previously considered "imaginary" or non-existent. The imaginary unit allows for the manipulation and calculation of complex numbers, which have both a real and imaginary component.

5. Sejam Z1 = 2x + yi e Z2 = 3i - 4. Se Z1 = Z2, então x + y é igual a:

0

1

2

3

If Z1 = Z2, then the real parts and imaginary parts of both complex numbers must be equal. From Z1 = 2x + yi, we can see that the real part is 2x and the imaginary part is y. From Z2 = 3i - 4, we can see that the real part is 0 and the imaginary part is -4. Therefore, we can equate the real parts and imaginary parts of Z1 and Z2 to get the following equations: 2x = 0 and y = -4. Solving the first equation, we find that x = 0. Substituting this value of x into the second equation, we find that y = -4. Therefore, x + y = 0 + (-4) = -4.

Explanation

If Z1 = Z2, then the real parts and imaginary parts of both complex numbers must be equal. From Z1 = 2x + yi, we can see that the real part is 2x and the imaginary part is y. From Z2 = 3i - 4, we can see that the real part is 0 and the imaginary part is -4. Therefore, we can equate the real parts and imaginary parts of Z1 and Z2 to get the following equations: 2x = 0 and y = -4. Solving the first equation, we find that x = 0. Substituting this value of x into the second equation, we find that y = -4. Therefore, x + y = 0 + (-4) = -4.

6. (1 + i)^2 = OBS: (a notação a^b equivale a "a" elevado a "b")

2

2i

0

2 + 2i

The expression (1 + i)^2 can be expanded using the FOIL method, which stands for First, Outer, Inner, Last.

First, we multiply the first terms: 1 * 1 = 1.
Outer, we multiply the outer terms: 1 * i = i.
Inner, we multiply the inner terms: i * 1 = i.
Last, we multiply the last terms: i * i = -1.

Combining all the terms, we have 1 + i + i - 1. The -1 and +1 cancel each other out, leaving us with 2i as the final answer.

Explanation

The expression (1 + i)^2 can be expanded using the FOIL method, which stands for First, Outer, Inner, Last.

First, we multiply the first terms: 1 * 1 = 1.
Outer, we multiply the outer terms: 1 * i = i.
Inner, we multiply the inner terms: i * 1 = i.
Last, we multiply the last terms: i * i = -1.

Combining all the terms, we have 1 + i + i - 1. The -1 and +1 cancel each other out, leaving us with 2i as the final answer.

7. Uma simplificação de i^162 pode ser: OBS: (a notação a^b equivale a "a" elevado a "b")

1

I

-1

-i

The given options represent the powers of the imaginary unit i. The power of i cycles in a pattern: i, -1, -i, 1. Since the power of i repeats every 4, we can simplify i^162 by dividing the exponent by 4. 162 divided by 4 gives a remainder of 2. Therefore, i^162 is equivalent to i^2, which is -1.

Explanation

The given options represent the powers of the imaginary unit i. The power of i cycles in a pattern: i, -1, -i, 1. Since the power of i repeats every 4, we can simplify i^162 by dividing the exponent by 4. 162 divided by 4 gives a remainder of 2. Therefore, i^162 is equivalent to i^2, which is -1.

8. O Matemático italiano que Publicou Ars Magna, obra na qual encontra-se a fórmula para resolução das equações de terceiro grau do tipo x^3 + px + q = 0, com p e q reais, foi OBS: (a notação a^b equivale a "a" elevado a "b")

Cardano

Tartaglia

Bombelli

Gorgonzzola

Cardano is the correct answer because he was the Italian mathematician who published the book "Ars Magna" which contains the formula for solving cubic equations of the form x^3 + px + q = 0. This formula is known as Cardano's formula and it revolutionized the field of algebra. Tartaglia and Bombelli were also important mathematicians of the time, but they did not publish the specific formula for solving cubic equations. Gorgonzzola is not a mathematician and is not relevant to the question.

Explanation

Cardano is the correct answer because he was the Italian mathematician who published the book "Ars Magna" which contains the formula for solving cubic equations of the form x^3 + px + q = 0. This formula is known as Cardano's formula and it revolutionized the field of algebra. Tartaglia and Bombelli were also important mathematicians of the time, but they did not publish the specific formula for solving cubic equations. Gorgonzzola is not a mathematician and is not relevant to the question.

9. (3 + 4i).(3 - 4i) =

25i

25

9 - 16i

- 7

The given expression is a multiplication of two complex numbers. When multiplying complex numbers, we use the distributive property and the fact that i^2 = -1. By multiplying (3 + 4i) and (3 - 4i), we get 9 - 12i + 12i - 16i^2. Simplifying further, we have 9 - 16i^2. Since i^2 = -1, this becomes 9 - 16(-1), which equals 9 + 16 = 25. Therefore, the correct answer is 25.

Explanation

The given expression is a multiplication of two complex numbers. When multiplying complex numbers, we use the distributive property and the fact that i^2 = -1. By multiplying (3 + 4i) and (3 - 4i), we get 9 - 12i + 12i - 16i^2. Simplifying further, we have 9 - 16i^2. Since i^2 = -1, this becomes 9 - 16(-1), which equals 9 + 16 = 25. Therefore, the correct answer is 25.

10. A parte imaginária do número complexo Z = a + bi é

A

B

Bi

I

The correct answer is b. In a complex number Z = a + bi, the imaginary part is represented by bi. In this case, the imaginary part of the complex number Z is bi.

Explanation

The correct answer is b. In a complex number Z = a + bi, the imaginary part is represented by bi. In this case, the imaginary part of the complex number Z is bi.

11. Leia as afirmações a seguir. I - Todo número real é complexo II - Todo número complexo é real III - Nenhum número imaginário é real

Estão corretas

Apenas I e II

Apenas I e III

Apenas II e III

I, II e III

The correct answer is "Apenas I e III" (Only I and III). This is because statement I is true, as every real number can be considered a complex number with an imaginary part of 0. Statement III is also true because imaginary numbers, by definition, are not real numbers. Statement II is incorrect because not all complex numbers are real numbers, as they can have a non-zero imaginary part.

Explanation

The correct answer is "Apenas I e III" (Only I and III). This is because statement I is true, as every real number can be considered a complex number with an imaginary part of 0. Statement III is also true because imaginary numbers, by definition, are not real numbers. Statement II is incorrect because not all complex numbers are real numbers, as they can have a non-zero imaginary part.

12. Se o número complexo Z = (x-1) + (y+2)i é imaginário puro, então (o símbolo <> significa DIFERENTE)

X = 1

Y = 2

Y = -2

X = 1 e y -2

If the complex number Z = (x-1) + (y+2)i is purely imaginary, it means that the real part of the complex number is equal to zero. In this case, the real part is (x-1). Since we want the real part to be zero, we can set (x-1) = 0, which means x = 1. Therefore, the correct answer is x = 1. As for y, there is no information given about its value, so we cannot determine its exact value.

Explanation

If the complex number Z = (x-1) + (y+2)i is purely imaginary, it means that the real part of the complex number is equal to zero. In this case, the real part is (x-1). Since we want the real part to be zero, we can set (x-1) = 0, which means x = 1. Therefore, the correct answer is x = 1. As for y, there is no information given about its value, so we cannot determine its exact value.

13. O problema: encontrar a medida x da aresta de um cubo, cujo volume seja 4 unidades maior que o volume de um paralelepípedo de área da base constante e igual a 15, e altura de medida x, pode ser modelado pela equação

X^3 - 15x - 4 = 0

X^3 = 15x - 4

X^3 = 4 - 15x

X = 15x + 4

The given equation x^3 - 15x - 4 = 0 represents the problem of finding the measure x of the edge of a cube, whose volume is 4 units larger than the volume of a rectangular parallelepiped with a constant base area of 15 and a height of measure x. The equation is obtained by setting the volume of the cube equal to the volume of the parallelepiped and rearranging the terms. The solution to this equation will give the value of x that satisfies the given conditions.

Explanation

The given equation x^3 - 15x - 4 = 0 represents the problem of finding the measure x of the edge of a cube, whose volume is 4 units larger than the volume of a rectangular parallelepiped with a constant base area of 15 and a height of measure x. The equation is obtained by setting the volume of the cube equal to the volume of the parallelepiped and rearranging the terms. The solution to this equation will give the value of x that satisfies the given conditions.

14. Nos números complexos, a raiz quadrada de -121 é

- 11

11i

+ 11 i ou - 11 i

Não existe raiz quadrada de numero negativo

In the realm of complex numbers, the square root of -121 is ±11i. This is because there is no real square root of a negative number, but in complex numbers, we use "i" to represent the imaginary unit. Therefore, the square root of -121 is ±11i, where "±" indicates that there are two possible solutions: 11i and -11i.

Explanation

In the realm of complex numbers, the square root of -121 is ±11i. This is because there is no real square root of a negative number, but in complex numbers, we use "i" to represent the imaginary unit. Therefore, the square root of -121 is ±11i, where "±" indicates that there are two possible solutions: 11i and -11i.

15. As raízes da equação x^2 + 1 = 0 são OBS: (a notação a^b equivale a "a" elevado a "b")

1 e -1

I

I e -i

Essa equação não tem raizes

The equation x^2 + 1 = 0 is a quadratic equation. In this equation, the coefficient of x^2 is 1, and the constant term is 1. When we solve this equation, we can use the quadratic formula, which states that the roots of the equation ax^2 + bx + c = 0 are given by the formula x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = 0, and c = 1. Plugging these values into the formula, we get x = ± √(-1), which simplifies to x = ± i. Therefore, the roots of the equation x^2 + 1 = 0 are i and -i.

Explanation

The equation x^2 + 1 = 0 is a quadratic equation. In this equation, the coefficient of x^2 is 1, and the constant term is 1. When we solve this equation, we can use the quadratic formula, which states that the roots of the equation ax^2 + bx + c = 0 are given by the formula x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = 0, and c = 1. Plugging these values into the formula, we get x = ± √(-1), which simplifies to x = ± i. Therefore, the roots of the equation x^2 + 1 = 0 are i and -i.