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The complex numbers are a field.

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Q: Are complex numbers under addition and multiplication a field?
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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).

Related questions

Which set of numbers forms a field with respect to the operations of addition and multiplication?

whole numbers

Does the set of irrational numbers with the usual addition and multiplication form a field?

Strictly speaking, no, because the identity for addition 0, and the identity for multiplication, 1 are not irrationals.

Properties of real number?

Real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced.

Expressions like 4(3 plus 2) and 4(3) plus 4(2)?

Such expressions illustrate the distributive property of multiplication over addition in the field of real numbers.

Should be distributive property of addition and multiplication have the same answer?

Yes, the Distributive Property is true over addition and multiplication, and it will continue to until you start studying exotic concepts such as Ring Theory or Field Theory.

What is the field of mathematics?

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

What is the difference between arithmetic and math?

Mathematics (math) is a broad field of endeavour, which includes arithmetic. Arithmetic is the part which deals with numbers (and their interactions) only. Other math fields are Number Theory, complex numbers, graph theory, differential calculus, many others.

What are the example of field that contains subfield?

The set of complex numbers is a field which contains the set of real numbers as a proper subfield.

How does a complex fraction differ from other types of fractions?

It does not, except that the fraction is a number in the complex field rather than real numbers.It does not, except that the fraction is a number in the complex field rather than real numbers.It does not, except that the fraction is a number in the complex field rather than real numbers.It does not, except that the fraction is a number in the complex field rather than real numbers.

What are complex decimals?

They are numbers in the complex field where the two components - the real and imaginary parts - are expressed in decimal form.

Is the set of positive integers a field under addition and multiplication?

NO. Certainly not. Additive inverse and Multiplicative inverse doesn't exist for many elements.

What does quaternion mean?

A quaternion is a mathematical concept in the field of algebra. It is an extension of the complex numbers and consists of four components: a real part and three imaginary parts. Quaternions are often used in computer graphics and robotics to represent three-dimensional rotations and orientations.