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What happens when you square root a negative number?
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.
The sum of two complex numbers is always a complex number?
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).
Why can't we add complex number in polar form?
You can, but the process is slightly complicated, because addition in the Complex field is like vector addition. If z1 = (r1, a1), and If z2 = (r2, a2) Then, if z = (r, a) r = sqrt(r12 + r22) and a = arctan[(r1sina1 + r2sina2)/(r1cosa1 + r2cosa2)]
What are the real number properties and the examples of it?
The set of real numbers, R, is a mathematical field. For any three real numbers x, y and z and the operations of addition and multiplication,• x + y belongs to R (closure under addition)• (x + y) + z = x + (y + z) (associative property of addition)• There is an element, 0, in R, such that x + 0 = 0 + x = x (existence of additive identity)• There is an element, -x, in R, such that x + (-x) = (-x) + x = 0 (existence of additive inverse)• x + y = y + x (Abelian or commutative property of addition)• x * y belongs to R (closure under multiplication)• (x * y) * z = x * (y * z) (associative property of multiplication)• There is an element, 1, in R, such that x * 1 = 1 * x = x (existence of multiplicative identity)• For every non-zero x, there is an element, 1/x, in R, such that x * 1/x = 1/x * x = 1 (existence of multiplicative inverse)• x * (y + z) = x*y + x * z (distributive property of multiplication over addition)
What is the significance of complex roots?
One significant feature of complex numbers is that all polynomial equations of order n, in the complex field, have n solutions. When multiple roots are Given any set of complex numbers {a(0),  … , a(n)}, such that at least one of a(1) to a(n) is non-zero, the equation a(n)*z^n + a(n-1)*z^(n-1) + ... + a(0) has at least one solution in the complex field. This is the Fundamental Theorem of Algebra and establishes the set of Complex numbers as a closed field. [a(0), ... , a(n) should be written with suffices but this browser has decided not to be cooperative!] The above solution is the complex root of the equation. In fact, if the equation is of order n, that is, if the coefficient a(n) is non-zero then, taking account of the multiplicity, the equation has exactly n roots (some of which may be real).
