- #1
- 5
- 0
does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
There is one similarity between natural numbers and complex numbers which is that two concecutive integers multiplied together then multiplied by 4 is one less than a perfect square. In other words if n is an integer then 4n(n+1)+1 is a perfect square. Likewise if z is a Gausian integer then 4z(z+1) + 1 is a perfect complex square. This leads me to define two consecutive Gausian Integers to be z and z plus 1.achyut joshi said:does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
I think Halls was referring to the property of being dense(http://en.wikipedia.org/wiki/Dense_set).disregardthat said:Sure they do. Every set can be well-ordered.
Give the complex numbers a well-ordering. For each complex number z, consider the minimal element of the set of complex numbers larger than z. This will be the least number larger than z, in other words, a consecutive element.
Now this of course applies to R and Q as well, but note that this cannot correspond to their usual order. The complex numbers however don't have any natural ordering (which satisfies e.g. a < b --> a+c < b+c), so I don't see how the order properties of the real numbers extends to the complex numbers.
achyut joshi said:does complex number have anything like consicutive numbers? . Like 2,3,4 are consicutive integers...
robert2734 said:suppose for numbers with the same norm, we arbitrarily list numbers counter clockwise from zero degrees. Then we can definitively say that one complex number comes before or after another in our catalog.
robert2734 said:suppose for numbers with the same norm, we arbitrarily list numbers counter clockwise from zero degrees. Then we can definitively say that one complex number comes before or after another in our catalog.
achyut joshi said:does complex number have anything like consicutive numbers? . Like 2,3,4 are consecutive integers...
robert2734 said:Consider the integers. We have -1< 0, -2< 0 and (-1)*(-2)>0. So the integers aren't an "orderred field" either but they do have an order to them. So back to OP's question: Can we order the complex numbers? yes. Does that order mean anything? I don't know.
Yes, complex numbers have a concept of consecutive numbers, but it is different from the concept of consecutive numbers in real numbers. In complex numbers, consecutive numbers are defined as numbers that are separated by a distance of 1 on the complex plane. This means that the real and imaginary parts of the numbers differ by 1.
Consecutive complex numbers are represented on the complex plane as points that are one unit apart on the x-axis. This means that the real parts of the numbers are consecutive, while the imaginary parts remain the same.
Yes, a complex number can have more than one consecutive number. In fact, every complex number has an infinite number of consecutive numbers, as there is an infinite number of points that are one unit apart on the complex plane.
Yes, there are patterns in the consecutive complex numbers. For example, the consecutive numbers of 3+4i are 4+4i, 5+4i, 6+4i, etc. This pattern can be seen on the complex plane as points forming a straight line parallel to the real axis.
Consecutive complex numbers are used in various areas of mathematics, including complex analysis, number theory, and algebra. They can be used to solve equations and problems involving complex numbers, and also play a role in the study of complex functions and polynomials.