I will give you brainliest, if you provide an accurate explanation.

Please include a thoughtful answer.

If not, I WILL report you.

1. Find the sqrt(i) and see if there is anything interesting that comes up.

2. Prove that complex numbers are closed under addition, subtraction, division, and cube roots


Answer 1


sqrt(i) =0.707106781 + 0.707106781 i

Most of the numbers we know, and work with, are Real Numbers. The Real Number System includes counting numbers, fractions, terminating decimals, positive numbers, negative numbers, zero, repeating decimals, never ending and non-repeating decimals, numbers that are expressed as radicals, and even pi (π).

The natural numbers are the set of counting numbers

• There are infinitely many numbers in a set of numbers.

• The natural numbers are "closed" under addition and multiplication.

The addition of two natural numbers creates another natural number.

The multiplication of two natural numbers creates another natural number.

closed under addition and multiplication.

BUT ...

The subtraction of two natural numbers does NOT necessarily create another natural number

The division of two natural numbers does NOT necessarily create another natural number  

The whole numbers are the set of counting numbers (natural numbers) along with zero


• There are infinitely many numbers in this set of numbers.

• The set of whole numbers is "closed" under addition and multiplication.


• The integers are the set of all of the natural numbers,

    plus their additive inverses and zero

• The integers are "closed" under addition, multiplication and subtraction,

    but NOT under division

Rational Numbers:

• The rational numbers are the set of numbers which can be expressed as a ratio

    (a fraction) between two integers.

• Integers are rational numbers since 5 can be written as the fraction 5/1.

• Decimals which terminate are rational numbers.

• Decimals which have a repeating pattern are rational numbers. 1/3 = 0.3333333...

• The rational numbers are "closed" under addition, subtraction, and multiplication. Under division, we run into the problem of division by 0, which makes the statement that "the rationals are closed under division" false. Some texts state that "the rationals are closed under division as long as the division is not by zero" which is a true statement.

Irrational Numbers:

The irrational numbers are the set of number which can NOT be written as a ratio (fraction).

• Decimals which never end nor repeat are irrational numbers.

• Irrational numbers are "not closed" under addition, subtraction, multiplication or division.

• Examples of irrational numbers: rad2, π

Answer 2

This is figured out by first figuring out that:

√(√(i)) = [tex]i^{1/4\\}[/tex]

Let [tex]\sqrt{i}[/tex] be a + b[tex]i[/tex].

Square both sides to get [tex]i[/tex] = (a + b[tex]i[/tex])(a + b[tex]i[/tex])

So, [tex]i[/tex] = a² - b² + 2ab[tex]i[/tex]

a² - b² is the real part. The real part is 0.

0 + [tex]i[/tex] = a² - b² + 2ab[tex]i[/tex]

a² - b² = 0

[tex]i[/tex] = 2ab[tex]i[/tex]

2ab = 1

ab = 1/2

b = 1/2a

Now, since a² - b² = 0,

a² - (1/2a)² = 0

a² - 1/(4a²) = 0

[tex]\frac{(4a^{4}-1 )}{4a^{2} }[/tex] = 0

That means the numerator MUST be 0.

[tex](4a^{4}-1 )}[/tex] = 0

[tex]4a^{4}[/tex] = 1

[tex]a^{4}[/tex] = 1/4

a = ± [tex]\frac{1}{\sqrt{2} }[/tex]

if a = [tex]\frac{1}{\sqrt{2} }[/tex] , then b = [tex]\frac{1}{\sqrt{2} }[/tex]

If a = -[tex]\frac{1}{\sqrt{2} }[/tex] , then b = -[tex]\frac{1}{\sqrt{2} }[/tex]

So, it's either a = -[tex]\frac{1}{\sqrt{2} }[/tex]  and b = -[tex]\frac{1}{\sqrt{2} }[/tex] , or a = [tex]\frac{1}{\sqrt{2} }[/tex]  and b = [tex]\frac{1}{\sqrt{2} }[/tex]

That's why sqrt(i) =0.707106781 + 0.707106781i, or sqrt(i) = -0.707106781 - 0.707106781i

Related Questions

Jay bought a painting on sale for 45% off the original price, for $27. What was the price of the painting before the sale?



 0.55x = $27

Step-by-step explanation:

Let the price of the painting = x

x - 45% of x = $27

    x - 0.45x = $27

         0.55x = $27

Divide both sides by 0.55

 0.55x/0.55 = $27/0.55

                  x = $49.09

1 less than the quotient of a number n and 6



[tex]\frac{n}{6}[/tex] - 1

Step-by-step explanation:

whenever the word than is there (less than, greater than) you have to switch the order so it the words 1 was first or in the beginning of the sentence but since the words less than are there, 1 goes toward the end or after the quotient of a number n and 6.

whats pi to the thousandth digit? no google or spam please and a explanation


Answer: PI to the thousandth digit is 8.

Step-by-step explanation: Since 9 is the last digit in PI, and 3 is the digit at the beginning of PI, then that means that 8 is PI to the thousandth digit.

How do u solve this question?


Op I don’t know, but to find the volume is length x width x height

NASCAR driver Jimmy Johnson completed 200 laps as he crossed the finish line. From the start of the race to the finish line, is a description of his _________________.

A. displacement
B. speed
C. velocity
D. Distance




Step-by-step explanation:

think its B im not sure

It would be d because the speed isnt said, displacement is for boats and velocity is basically speed and its how many laps, so it is distance


p: you pass the class

q: you go to the next grade

p -> ~q:_______________________________




P is you pass the class, so it's the hypothesis in the conditional. And Q is the conclusion, which ic you got to the next grade.

Step-by-step explanation:

In order to got to the next grade, you need to pass the class.


If you pass the class, you go to the next grade.

What is the slope of the line y = -2?




Step-by-step explanation:

There is no x, therefore there is no slope.

Find the distance between the points
(9,-7) and





Step-by-step explanation:

5 is the 100% correct answer.....Hopes to get BRAINLIEST.....

Thank you ☺️☺️

please help. you can select a,b,c,d or multiple.


Answer: b & d

Step-by-step explanation:

These are complex roots because they do not have x-intercepts.

Hope this helps!

if 3n-4=8 then what is the value of 5n-3 ​




Step-by-step explanation:


3n - 4 = 8 ( add 4 to both sides )

3n = 12 ( divide both sides by 3 )

n = 4


5n - 3 = 5(4) - 3 = 20 - 3 = 17

I should pay attention in class more smh



I think the slope is unidentified.

Hope this Helps!

There are 84 turkey and 63 ham sandwiches to be places on trays. Each tray should have only one kind of sandwich, and all trays have the same number of sandwiches. If you place the greatest possible number of sandwiches on each tray, how many trays will you have with turkey and how many with ham?


it should have 83 turkey and 83 ham and one extra.




Step-by-step explanation:

Ist method - Area of the triangle + Area of the  trapezium

Area of triangle = 1/2 × b × h

here, b = 8 cm but h is not given

so we will find h by pythagoras theorem

AC^2 = AB^2 + BC^2

10^2 = 8^2 + BC^2

100 = 64 + BC^2

BC^2 = 36

BC = √36

BC = 6

∴ h = 6cm

area of triangle = 1/2 × b × h

= 1/2 × 8 × 6

= 24 cm^2

Area of trapezium = 1/2 ( a + b ) h

Here a = 13cm

         b = 25 - 8 = 17 cm

         h = 6 cm

So area = 1/2 ( a + b ) h

= 1/2 ( 13 + 17 ) 6

= 90 cm^2

So total area =  Area of the triangle + Area of the  trapezium

= 24 cm^2 + 90 cm^2 = 114 cm^2

2nd method - Area of the entire figure

Since the figure is a trapezium,

Area of trapezium = 1/2 ( a + b ) h

here, a = 13cm

        b = 25 cm

        h = 6 cm

Area of trapezium = 1/2 ( a + b ) h

= 1/2 ( 13 + 25 ) 6

= 114 cm^2

Hope this helps

plz mark as brainliest!!!!!!!!!!


not a very clear diagram sorry luv but i just cant really see the question  <3

step-by-step explanation:

What is the missing term in the factorization? 1872 -- 32 2(3x+?) (3x – 4) Enter your answer in the box.​




Step-by-step explanation:


1-cos2a/1+cosa if 1/3 how much sin²a



(1-cos2A) /(1+cos2A) =tan²A


We know that,

cos(A+B) =cosA.cosB-sinA.sinB


=>cos2A=cosA.cosA - sinA.sinA



Since {cos²A+sin²A=1}

Divide the numerator & the denominator by (cos²A) to get,

cos2A = {(cos²A-sin²A) ÷cos²A} / {(cos²A+sin²A) ÷cos²A}

cos2A ={(1-tan²A)/(1+tan²A)}


1-cos2A = 1-[{(1–tan²A)/(1+tan²A)}]

1-cos2A =(1+tan²A-1+tan²A)/(1+tan²A)


And now.......




So now,

(1-cos2A)/(1+cos2A)= {2tan²A/(1+tan²A)}÷{2/(1+tan²A)}



Step-by-step explanation:

make me as brain liest

The total amount of Keira's cell phone bill, b, varies directly with the number of text messages she sends, m. Which of the following equations best models the amount of money Keira must pay for her cell phone bill?


Pls list the equation options so we can give you an answer

I have 2 questions
(1) What is the value of x?
(2) Classify the Triangle by its Angles. HINT: Use the Triangle Sum Theorem to find out the measure of the angles.


The first triangle: x=25 and is a Acute angle

The second triangle: x=30 and is a right triangle

The perimeter of a rectangle is 72 inches. The ratio of the width to the length is 4:5. What is the width of the rectangle?


9514 1404 393


  16 inches

Step-by-step explanation:

The sum of length and width is half the perimeter, so is 36 inches.

The width is 4/(4+5) = 4/9 of the total of length and width, so is ...

  (4/9)(36 inches) = 16 inches

The width of the rectangle is 16 inches.

What is the measure of angle F?

A. Cannot be determined
B. 65°
C. 55°
D. 75°



65 degrees since both opposite sides are 10


65 degrees

Step-by-step explanation:

Just took the test

In an electrical circuit, the voltage across a resistor is directly proportional to the current running through the resistor. If a current of 12 amps produces 480 volts across a resistor, how many volts would a current of 1.5 amps produce across an identical resistor? Record your answer as number only.


Answer:volts =60v

Step-by-step explanation:


 The voltage across a resistor is directly proportional to the current running through the resistor Is mathematically represented as

V∝ C

introducing the proportion of constant , we have


Step 2

When current =12 amps, and volts = 480, we have


480=K  X  12

Constant of proportionality ,K= V/C=480/12= 40

Therefore, when current = 1.5 amp, volts=?


V= 40  X 1.5

V= 60 volts

I need help in this before 11:59 please


Step-by-step explanation:

√7 * √7 = 7 , which is not a irrational number

Hannah uses a store gift card to pay for 3 pair of pants priced at $27 each. What is the change in
the value of the store gift card?



If the gift card is $30 the remaining change is 3 dollars

Can function notations be simplified?
I was working on this equation: f(x) = -6/9x + 300
I was wondering if I can convert the fraction ‘-6/9’ into ‘-2/3’?


Step-by-step explanation:

it's not same but u can find your answer through it

What does 7 hundreds×10 equal?


Answer:700x 10= 7,000

Answer: 7000

Step-by-step explanation:

0.3 km are in how many miles?



There are 0.186411 miles in 0.3 km.

0.3 km = 0.19 miles hmm

Oscar is raising monarch caterpillars for a summer science project. Every day, he feeds 4 fresh milkweed leaves to each caterpillar. Oscar wants to know how many caterpillars he can feed in a day with a total of 28 leaves. Which equation can Oscar use to find n, the number of caterpillars he can feed in a day?


Answer: n = 28/4

Step-by-step explanation:

Number of leaves uses to feed each caterpillar = 4.

If Oscar has 28 leaves, the number of caterpillars n he'll be able to feed in a day will be represented by the equation below:

n = 28/4

This implies that 7 caterpillars can be fed in a day.

0.333 repeating times 15. Please help me out and answer this question.




Step-by-step explanation:

The members of the Math team contributed a total of $9.10 for the party after their practice. Each member contributed the same amount of money in dimes only. How many members are in the team and how much money did everyone contribute? Give all possible answers.



Number of possibilities

1) 91 members that contribute 1 dime each ($0.10)

2) 13 members that contribute 7 dimes each ($0.10 × 7 = $0.70)

3) 70 members that contribute 13 dimes each ($0.10 × 13 = $0.13)

Step-by-step explanation:

A dime = $0.10 in dollars

We are told that:

The total amount contributed = $9.10 for the party after their practice.

Each member contributed the same amount of money in dimes only.

Let the number of members be represented as: x

Number of possibilities

We find the factors of $9.1

= 1, 0.7, 0.13, 9.1

= 1 × 9.1

= 0.7 × 13

= 0.13 × 70


1) $0.10 × x = $9.10

x = $9.10/$0.10

x = 91

Therefore, from the calculation above, there are 91 members and each of them contributed the same amount which was 1 dime ($0.10).


7($0.10 )× x = $9.10

x = $9.10/$0.70

x = 13

Therefore, from the calculation above, there are 13 members that contribute 7 dimes each ($0.10 × 7 = $0.70)

3) 13($0.10) × x = $9.10

x = $9.10/$0.13

x = 70

Therefore, from the calculation above, there are 70 members that contribute 13 dimes each ($0.10 × 13 = $0.13)

The graph shows the cost of purchasing T-shirts. Find the constant rate of change for the graph. Then explain what points (0,0) and (1,9) represent​



See below

Step-by-step explanation:

As per the attached graph we observe that the constant rate of change is 9 as each step of horizontal change corresponds to 9 steps of vertical change

It is found as the division below:

(9-0)/(1-0) = 9/1 = 9

Point (0, 0) indicates the start point

Point (1, 9) indicates the cost of 1 t-shirt


As to the attachment graph ;

we observed that the constant rate is 9each step of horizontal change correspond to 9 steps of vertical change


according to the question

it is found as the division below ;

(9-0)/(1-0)=9/1 = 9

the point which is indicating the starting point ;


the point which indicate the cost of one t-shirt ;






Step-by-step explanation:

Other Questions
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