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The atomic number is a fundamental characteristic of an element, denoting the number of protons present in the nucleus of an atom. It is represented by the symbol 'Z' and is crucial because it defines the chemical identity of an atom. For instance, all carbon atoms have an atomic number of 6, which means every carbon atom contains 6 protons in its nucleus.

Understanding the atomic number is essential for categorizing elements in the periodic table and for predicting how elements will react chemically. Since the atomic number is directly related to the number of protons, it also indirectly influences the number of electrons, given that a neutral atom has equal numbers of protons and electrons. This balance determines an atom's electrical neutrality and its ability to bond with other atoms.

Isotopes are variants of a particular chemical element which differ in neutron number, although having the same number of protons, thus sharing the same atomic number. For example, the isotopes of hydrogen, the most abundant element in the universe, are protium with 0 neutrons, deuterium with 1 neutron, and tritium with 2 neutrons. Each of these isotopes of hydrogen has an atomic number of 1, indicating a single proton in the nucleus, but their differing neutron count makes their atomic masses vary.

Neutrons play a critical role in the stability of an atom. While changing the number of protons transforms the element, altering the neutrons leads to different isotopes which can vary in stability. Some isotopes with too many or too few neutrons compared to protons are unstable and can lead to radioactive decay, crucial in the process of radiometric dating and other scientific applications.

Radioisotopes, or radioactive isotopes, are variants of an element that have unstable combinations of protons and neutrons, leading to radioactive decay. This decay process involves the emission of radiation in the form of particles or electromagnetic waves, which can be utilized for various applications ranging from medical treatments to power generation.

For instance, Carbon-14 is a radioisotope of carbon that is used in carbon dating to determine the age of archaeological artifacts. In medicine, radioisotopes like Iodine-131 are used in the diagnosis and treatment of thyroid conditions. These isotopes decay over time, which means they lose energy by emitting radiation, and this very property is exploited in various fields to detect, diagnose, and treat diseases, as well as in industrial applications such as non-destructive testing of materials.