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Q: Why do principal roots of negative numbers have to be different in real numbers or complex numbers?

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Probably because if you consider real numbers, you are not interested in complex numbers.Any complex number other than zero - and that includes real numbers - has three cubic roots, which have an angle of 120 degrees between one another. For example, the cubic roots of 1 are 1, 1 at an angle of 120°, and 1 at an angle of 240°. Similarly, the cubic roots of -1 are 1 at an angle of 180° (equal to -1), 1 at an angle of 60°, and 1 at an angle of 300°.

Complex numbers are the square roots of negative numbers. i.e. root -1 = i

You can, if you consider complex numbers.

Obviously, there are an infinite number of ways you can classify numbers.For example, you can classify positive and negative numbers; integers and non-integers; rational and irratinoal numbers; real numbers and complex numbers.Obviously, there are an infinite number of ways you can classify numbers.For example, you can classify positive and negative numbers; integers and non-integers; rational and irratinoal numbers; real numbers and complex numbers.Obviously, there are an infinite number of ways you can classify numbers.For example, you can classify positive and negative numbers; integers and non-integers; rational and irratinoal numbers; real numbers and complex numbers.Obviously, there are an infinite number of ways you can classify numbers.For example, you can classify positive and negative numbers; integers and non-integers; rational and irratinoal numbers; real numbers and complex numbers.

No. Negative numbers form a set of numbers. Numbers written as decimals is a notation for writing numbers. Different things.Negative numbers can be written as decimals, fractions, complex numbers, etc.Numbers written as decimals can be either negative numbers, zero, or positive numbers. They can also be integer numbers or real numbers.

The square root of a negative number is not real. However, there is a field of numbers known as the complex number field which contains the reals and in which negative numbers have square roots. Complex numbers can all be expressed in the form a+bi where a and b are real and i is the pure imaginary such that i2=1. Please see the related links for more information about complex and imaginary numbers.

Non-Zero Real Numbers are infact complex conjugate numbers. They are negative prime numbers.

The square of a "normal" number is not negative. Consequently, within real numbers, the square root of a negative number cannot exist. However, they do exist within complex numbers (which include real numbers)and, if you do study the theory of complex numbers you wil find that all the familiar properties are true.

Numbers that can be positive or negative include the integers, the rational numbers, the real numbers, and the complex numbers. All integers are rational numbers (numbers that can be written as a fraction, like 2/1), but most rational numbers are not integers -- like -1/2. (2/1, a rational, can be written as 2, an integer). The real numbers include all the rationals, plus many, many more numbers that can't be written as ratios or fractions, such as the square root of 2, pi, and the euler constant, e. As with the rational numbers and integers, there are as many negative real numbers as there are positive ones. Finally, we have the complex numbers. These include all of the real numbers, plus the roots of negative real numbers. Complex numbers are written in two parts -- a real part, plus an "imaginary" part (which is just as "real" as the real part, but is called "imaginary" for historical reasons). For example, 1 + i is a complex number with positive real and imaginary parts, while -1 - i is a complex number with negative real and imaginary parts. Positive and negative number systems are clearly very important in mathematics and in everyday life. They are all distinguished by the fact that they include magnitudes less than zero, as well as greater than zero (magnitudes of complex numbers are more complicated because complex numbers can have both positive and negative parts in one complex number!) There is also the term "non-zero" which refers to values that are positive or negative but not a value that is neither. It is a very important mathematical term since many functions (reciprocals, for example) can only have non-zero domains.

And its negative counterpart.

No. Negative four is a real number. All real numbers are also complex numbers, so it is a complex number (but it's real, not nonreal)

No, it is not irrational because it is a square root of a negative number - which falls into the set of Complex numbers. Irrational numbers can not have an imaginary component.

They are called real numbers. Negative square roots must be complex numbers.

The set of integers, the set of rational numbers, the set of real numbers, the set of complex numbers, ...

Negative rational numbers; Negative real numbers; Rational numbers; Real numbers. The number also belongs to the set of complex numbers, quaternions and supersets.

For most school mathematics, negative numbers do not have square roots. This is because a negative number multiplied by itself is a negative times a negative and so is positive. When (if) you study advanced mathematics, you will learn that there is a solution and this falls within the realms of complex mathematics and imaginary numbers.

Yes, for example in complex numbers z = 1 -i, i is the "x-component" and here it is negative.

The product of negative number and a positive number is always a negative. The product of two positive numbers, or two negative numbers, is always a positive.

The real numbers together with the imaginary numbers form a sort of vector. What these complex numbers (complex means the combination of real and imaginary numbers) represent depends on the specific situation. Just as there are situations where it doesn't make sense to use negative numbers, or fractional numbers, in many common situations it doesn't make sense to use complex numbers. In an electrical circuit (specifically, AC), the real numbers might represent resistance, while the imaginary number represent reactance - and voltages and currents are also represented by complex numbers, with the angle of the complex number representing how much one quantity is ahead or behind another quantity (the "phase angle"). In quantum mechanics, the complex numbers might not represent anything (perhaps they don't, I am not sure...), but they are useful for calculations.

A "complex number" is a number of the form a+bi, where a and b are both real numbers and i is the principal square root of -1. Since b can be equal to 0, you see that the real numbers are a subset of the complex numbers. Similarly, since a can be zero, the imaginary numbers are a subset of the complex numbers. So let's take two complex numbers: a+bi and c+di (where a, b, c, and d are real). We add them together and we get: (a+c) + (b+d)i The sum of two real numbers is always real, so a+c is a real number and b+d is a real number, so the sum of two complex numbers is a complex number. What you may really be wondering is whether the sum of two non-real complex numbers can ever be a real number. The answer is yes: (3+2i) + (5-2i) = 8. In fact, the complex numbers form an algebraic field. The sum, difference, product, and quotient of any two complex numbers (except division by 0) is a complex number (keeping in mind the special case that both real and imaginary numbers are a subset of the complex numbers).

This probably refers to how to handle computations with the set of Complex Numbers (which is a combination of the set of real numbers and imaginary numbers), rather than just complicatedcalculations, or calculations which are very involved and as-such appear very complex (which is a different thing than Complex Numbers).

Different whole numbers are always whole numbers, but I suspect you meant to ask about the difference between whole numbers. You can subtract two whole numbers and get a negative result. Whole numbers can't be negative.

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Yes. But until you study complex numbers, there is no solution.

If the signs of both numbers are the same, the product will be positive. If the signs of the numbers are different, the product will be negative.