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Closure: The sum of two real numbers is always a real number. Associativity: If a,b ,c are real numbers, then (a+b)+c = a+(b+c) Identity: 0 is the identity element since 0+a=a and a+0=a for any real number a. Inverse: Every real number (a) has an additive inverse (-a) since a + (-a) = 0 Those are the four requirements for a group.
2a + 6b = c, where a, b, and c are all numbers from the domain of your choice. Usually, in algebra, the real number set R is used as the domain. In this case, you would need to select values from R for a and b which will yield the value of c. Since this function is closed in the real number system, c will always be a real number, given that a and b are real numbers.
There is a hierarchy of sets:Counting numbersNatural numbers (N)Integers (Z)Rational numbers (Q)Real numbers (R)Complex numbers (C)Quaternions, and possibly more.
In (a+bi) + (c+di), you add the real parts using the laws for real numbers and do the same for the imanginary parts. (a+c)+(b+d)i
Blob
If you are talking about pure imaginary numbers (a complex number with no real part) then no. Example: bi times ci where b and c are real numbers equals b*c*i² = b*c*(-1) = -b*c, which is a real number, because b & c & -1 are all real numbers. If you're talking about multiplying two complex numbers (a + bi)*(c + di), then the product will be complex, but it could be real or imaginary, depending on the values of a, b, c, & d.
ab x ac = ab+c, where a, b, and c are all real numbers.
Closure: The sum of two real numbers is always a real number. Associativity: If a,b ,c are real numbers, then (a+b)+c = a+(b+c) Identity: 0 is the identity element since 0+a=a and a+0=a for any real number a. Inverse: Every real number (a) has an additive inverse (-a) since a + (-a) = 0 Those are the four requirements for a group.
Decimal numbers are real numbers. In C and C++ we use the float, double and long double data types to represent real numbers.
2a + 6b = c, where a, b, and c are all numbers from the domain of your choice. Usually, in algebra, the real number set R is used as the domain. In this case, you would need to select values from R for a and b which will yield the value of c. Since this function is closed in the real number system, c will always be a real number, given that a and b are real numbers.
There is a hierarchy of sets:Counting numbersNatural numbers (N)Integers (Z)Rational numbers (Q)Real numbers (R)Complex numbers (C)Quaternions, and possibly more.
In (a+bi) + (c+di), you add the real parts using the laws for real numbers and do the same for the imanginary parts. (a+c)+(b+d)i
The property which states that for all real numbers a, b, and c, their sum is always the same, regardless of their grouping:(a + b) + c = a + (b + c)
Real numbers are a proper subset of complex numbers. In fact each complex number, z, can be represented as z = x +iy where x and y are real numbers and i is the imaginary square root of -1.Thus the set of complex numbers is the Cartesian product of two sets of real numbers. That is, C = R x R where C is the set of complex numbers and R is the set of real numbers. Limitations of this browser prevent me from writing that in a mathematically precise and more helpful fashion.
To show that the set of all real numbers is an abelian group with respect to addition, we need to verify the group properties: Closure: For any two real numbers a and b, their sum a + b is also a real number. Associativity: Addition of real numbers is associative, meaning (a + b) + c = a + (b + c) for all real numbers a, b, and c. Identity element: The real number 0 serves as the identity element since a + 0 = a for all real numbers a. Inverse element: For every real number a, its additive inverse -a exists such that a + (-a) = 0. Commutativity: Addition of real numbers is commutative, meaning a + b = b + a for all real numbers a and b. Since the set of real numbers satisfies all these properties, it is indeed an abelian group with respect to addition.
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).