*R = real numbers, Z = integers, N = natural numbers, Q = rational numbers, and P = irrational numbers. There are also other sets of numbers used in number theory that overlap with some of these sets; see below for a list of them.
Number theorists study questions about these sets of numbers that arise in different contexts in mathematics. They often try to determine which properties each set of numbers has, and how the sets interact with one another. For example, many mathematicians wonder what properties each of the above sets of numbers has as a function of the other sets: Are there any relations between R, Z, N, or Q and P? Do all pairs of elements of R intersect, cover R, etc.? What about pairs of elements of Z, what can you say about them? Triples, quadruplets, etc.?
The study of numbers began with attempts by ancient Greeks to classify and order them. Modern number theory begins with Isaac Newton's work on calculus in the 17th century, which led to further advances in analysis made by Leibniz, Bernoulli, and others. In the 19th century, two German mathematicians, Carl Friedrich Gauss and Johann Carl Friedrich Abel, developed much of what is now known about prime numbers.
In mathematics, the letters R, Q, N, and Z stand for real numbers, rational numbers, natural numbers, and integers, respectively. Thus, all the above-mentioned numbers are included in the algebraic structure called "the real number field". The set of all real numbers along with addition and multiplication is called the real number line or simply the real line.
The letter Z also has other meanings in mathematics. It is used as a symbol for the complex plane and for the result of an integer division operation. When dividing two integers, if there is a remainder, then that remainder is added to the result of the first division; otherwise, it is assumed that the result of the division was zero. For example, if 12/4 equals 3 and not 3/1, then 12/4 = 3+1=4.
Thus, z is used as a placeholder value in cases where no value is specified. For example, z2 would be interpreted as any number multiplied by itself, whereas z3 would be interpreted as a number times a number times a number... You get the picture!
In mathematics, physics, and engineering, Z is often used as a symbol for impedance.
The symbol Z represents the set of integers and is derived from the German term Zahlen, which means "numbers." Positive integers are those that are strictly greater than zero, whereas negative integers are those that are strictly smaller than zero. The set of all positive and negative integers is called the full integer span or simply the integers.
Integers can be divided by the symbol "/", with the integer on the left being the divisor and the integer on the right being the dividend. For example, 12 / 3 is 4, and 92 / 7 is 14. Integer division always produces an answer with an exact number of digits after the decimal point. If there are fractions remaining, they must be rounded down in order to produce a whole number. For example, if 12/3 = 4, then 24/4 = 6; however, 21/4 = 5.5. Five and six are different numbers with four digits each, so they cannot be considered as equal.
Integers are important in mathematics because many mathematical concepts can be defined in terms of integers. For example, addition, subtraction, multiplication, and division of integers define a commutative ring with integers as its elements. There are also other rings known as integral domains where these operations are not only defined but also have unique solutions for any inputs.
The symbol for atomic number Z is "Zahl," which translates to "number" in German. Prior to 1915, the sign Z represented an element's position in the periodic table. When it was discovered that this was also the charge of the atom, Z was dubbed "Atomzahl," or atomic number.
When this response was acknowledged, the loading process began. Most commonly, Zn is used to represent the integers modulo n, which are represented as Zn = 0, 1, 2, n-1, the non-negative integers smaller than n. That is, if n is 23, then Zn represents the remainder when n is divided by 23.