# What is Integers: Definition and 472 Discussions

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold

(

Z

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{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

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1. ### Understanding the given proof of integers - Ring theory

My interest is on the highlighted part ... Now to my question, in what cases do we have ##mn<(m,n)[m,n]?## I was able to use my example say, Let ##m=24## and ##n=30## for example, then ##[m,n]=120## and ##(m,n)=6## in this case we can verify that, ##720=6⋅120## implying that, ##mn≤...
2. ### Find the GCD of the given complex numbers (Gaussian Integers)

Hello guys, I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean? ##z=1.2-1.4i## is rounded down to ##z=1-i##? I can see from here they came up with simultaneous equation i.e ##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
3. ### I Is this the correct way to quantify these integers?

Does the above quantifier represent/symbolize that all of the integers ## a, b, c, d ## cannot be ## 0 ##? Is this correct?
4. ### I Planck’s constant: Why are atom vibration frequencies integers only?

Given the definition of whole numbers as integers, https://www.google.com/search?q=what+is+a+whole+number&rlz=1C1VDKB_en-GB&oq=what+is+a+whole+number&aqs=chrome..69i57j0i512l9.11619j0j15&sourceid=chrome&ie=UTF-8 Is it known why atom vibrations are only at whole numbers ( ref plank’s constant)...

49. ### B Find triangles with areas that are integers

It is pretty obvious that all right-angled triangles whose sides are integers will have areas which are also integers. Since either the base or height will be an even number, half base x height will always come out exactly. However, I have only found one non-right-angled triangle where this is...
50. ### MHB Expressing First 1000 Positive Integers as Floor Functions

How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor+\lfloor 4x \rfloor+\lfloor 6x \rfloor+\lfloor 8x \rfloor$, where $x$ is a real number?