What does Modulus mean in Math? How can one work with it? Explain with the help of suitable examples.
The absolute value or modulus | x | of a real number x, is the non-negative value of x with disregard to the sign. Basically, | x | = x for a positive value of x, | x | = - x , for a negative value of x, where x is positive and | 0 | = 0. Example can be cited, the absolute value of 4 is 4 and absolute value of – 4 is also 4. The absolute value of a number is the distance between that number and zero. The generalization of absolute value of real numbers takes place in a wide variety of mathematical settings. For example, absolute value can be defined for the complex numbers, ordered rings, quaternions, vector spaces and fields. Absolute value is closely related with the notion of magnitude, norm and distance in different mathematical and physical contexts.
Any real number x, the absolute value or modulus of x can be denoted by | x | , represented using a vertical bar on either side of the quantity, and defined as
| x | = x, if x ≥ 0
| x | = - x if x < 0
The absolute value of x is always either positive or zero, but it is never negative, when x itself is negative ( x < 0 ), then the absolute value is in general positive ( | x | = - x > 0). The analytical geometric point of view, the absolute value of a real number is the distance between that number and zero along the line of real numbers, and more generalized purview of an absolute value of the difference between two real numbers is the actual distance between them. The notion of abstract distant function in mathematics is seen as a generalized view of the absolute value of the difference. The square root symbol represents an unique positive square root, when it is applied to a positive number, it is followed as | x | =√x2 , which is equivalent to the above definition and can be used as an alternative definition of the absolute value of a real number. The absolute value or modulus has four fundamental properties ( a , b considered to be real numbers ), which are used to provide a generalized purview of this notion to other domains
| a | ≥ 0 , non- negativity
| a | = 0 , positive definiteness
| ab | = | a | | b | , Multiplicavility
| a + b | ≤ | a | + | b | , Subadditivity, specifically the triangle inequality.
Non-negativity , positive definiteness and multiplicativity are apparent from the definition. Checking whether subadditivity holds, it is required to analyze one of the two alternatives of taking s, the value as either – 1 or + 1, which guarantees that s. ( a + b ) = | a + b | ≥ 0. Now, because -1.x ≤ | x | and + 1 . x ≤ | x |, it follows that, whichever is the value of s, one has s .x ≤ | x | for all real x, subsequently | a + b | = s. ( a + b ) = s. a + s. b ≤ | a | + | b | as required. To generalize the argument to complex numbers, it is required to see the proof of triangular inequality for complex numbers.
Modulo, often it is abbreviated as “mod” is a mathematical operation. This is like a division problem, except the answer is the remainder of an integer division operation, instead of a decimal result. Further illustration provided a decimal division, modulus is not looking for the result of the division. Modulus can be defined as a function in mathematics, that is capable of changing negative numbers to a positive number. Examples like | -3 | = 3 , | - 5 | = 5, however there is no change in positive numbers to negative numbers.
The real value of absolute value functions is continuous everywhere. This is differentiable everywhere except for when x = 0. This is monotonically decreasing at the interval and monotonically increasing on the interval, since a real number and the opposite number also has the same absolute value, this is an even function and thus not invertible. The real absolute value or modulus function is a piecewise linear, convex function.
The definition of absolute value given in case of real numbers can be extended to any ordered ring. If a is an element in an ordered ring R, then the absolute value of a, is denoted using | a | can be defined as
| a | = a, if a ≥ 0
| a | = - a, if a < 0,
Where, - a is an additive inverse of a, 0 is the additive identity element and < and ≥ have the same meaning with reference to the ordering in the ring.
The four fundamental properties of the absolute value in case of real numbers are used in generalization, where the notion of absolute value to an arbitrary field, is as follows. Real valued function v on a field F is termed as an absolute value, also modulus, magnitude, value or valuation. The absolute values for real numbers is used with a minimal modification to generalize the notion for an arbitrary vector space. Real value function on a vector space V, over a field F, which is reperesented as || . || , is called absolute value, but more usually a norm.
Compositional algebra A, has an involution x tends to x* called as the conjugation. The product of A is an element and the conjugate x* is written as N(X) = xx* and called the norm of x. The real number R, complex numbers C and quaternions H, are all composition algebras with norms given by definite quadratic forms. The absolute value used in division algebras is given by the square root of the composition algebra form.
MyAssignmenthelp.com has proved to be the best cheap assignment help provider in USA. Despite keeping the prices to a minimum, we keep providing top-notch quality college paper writing services for all types of academic papers. Our wide range of services includes report writing help, help with essays, dissertation writing services, coursework help. Students from all corners of the USA prefer to pay our experts for writing top quality assignments. Moreover, it is guaranteed they won't regret for paying our experts to write assignment.
Just share requirement and get customized Solution.
Orders
Overall Rating
Experts
Our writers make sure that all orders are submitted, prior to the deadline.
Using reliable plagiarism detection software, Turnitin.com.We only provide customized 100 percent original papers.
Feel free to contact our assignment writing services any time via phone, email or live chat. If you are unable to calculate word count online, ask our customer executives.
Our writers can provide you professional writing assistance on any subject at any level.
Our best price guarantee ensures that the features we offer cannot be matched by any of the competitors.
Get all your documents checked for plagiarism or duplicacy with us.
Get different kinds of essays typed in minutes with clicks.
Calculate your semester grades and cumulative GPa with our GPA Calculator.
Balance any chemical equation in minutes just by entering the formula.
Calculate the number of words and number of pages of all your academic documents.
Our Mission Client Satisfaction
it was really good and i will recomend it to my friends. i got above average marks for it
Australia
Dear Sir/Madam, Thank you very much for your expertise and assistance. I was very informative and very helpful. Regards,
Australia
Dear Sir/Madam, Thank you very much for the excellent work! It really helped me a lot. Regards,
Australia
the best team ever and very timely manners with affordable pricing and won’t have any complaints
Australia