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:

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.

Integers:

• 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

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

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