The triangle inequality is a fundamental property of algebraic operations on real numbers. It states that for any three real numbers a, b, and c, the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the remaining side. Symbolically, the triangle inequality can be…
The modulus and principal argument are two important properties of complex numbers. Here’s a brief explanation of each: Modulus: The modulus of a complex number z is defined as the distance between the origin and the point representing z in the complex plane. It is denoted by |z|. The modulus of a complex number can…
In algebra, polar representation refers to the representation of complex numbers in terms of their magnitude and angle. A complex number can be represented in polar form as: z = r(cosθ + i sinθ) where z is the complex number, r is its magnitude (or modulus), and θ is its angle (or argument). The angle…
Algebra conjugation is a mathematical operation that involves changing the sign of the imaginary part of a complex number. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i.e., i^2 = -1). The conjugate of…
Algebra multiplication involves multiplying variables and/or constants together. The general format for algebra multiplication is: a * b where “a” and “b” can be any combination of numbers or variables. For example: 2 * 3 = 6 x * y = xy When multiplying variables together, we can use the rules of exponents to simplify…
Algebra addition refers to the mathematical operation of combining two or more numbers or variables to get a sum. In algebra, addition is usually denoted by the plus sign (+). For example, if we have two variables, x and y, we can add them together using the expression “x + y”. The addition of numbers…
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as i^2 = -1. The algebraic operations on complex numbers are similar to those of real numbers. We can add, subtract, multiply and divide complex numbers. Addition: To add…
Product of Functions: The product of two functions f(x) and g(x) is denoted by f(x) * g(x) and is defined as (f * g)(x) = f(x) * g(x) for all x in the domain of both f and g. For example, if f(x) = x^2 and g(x) = sin(x), then (f * g)(x) = x^2…
The greatest integer function, denoted by ⌊x⌋ or sometimes by [x], is a mathematical function that returns the largest integer that is less than or equal to its argument x. For example, ⌊4.5⌋ = 4 and ⌊-2.7⌋ = -3. Formally, the greatest integer function is defined as follows: For any real number x, let n…
The absolute value of a real number is the distance of the number from zero on the number line. It is denoted by two vertical bars surrounding the number, like |x|. More formally, the absolute value of a real number x is defined as: |x| = x, if x is greater than or equal to…