Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today.
Observational error is the difference between a measured value of a quantity and its true value. In statistics, an error is not necessarily a "mistake". Variability is an inherent part of the results of measurements and of the measurement process.
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5".
Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law, is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as:
The absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system.
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:
In the field of psychology, cognitive dissonance is the perception of contradictory information and the mental toll of it. Relevant items of information include a person's actions, feelings, ideas, beliefs, values, and things in the environment. Cognitive dissonance is typically experienced as psychological stress when persons participate in an action that goes against one or more of those things. According to this theory, when two actions or ideas are not psychologically consistent with each other, people do all in their power to change them until they become consistent. The discomfort is triggered by the person's belief clashing with new information perceived, wherein the individual tries to find a way to resolve the contradiction to reduce their discomfort.
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.
In mathematics, −1 is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.
Elementary mathematics, also known as primary or secondary school school mathematics, is the study of mathematics topics that are commonly taught at the primary or secondary school levels around the world. It includes a wide range of mathematical concepts and skills, including number sense, algebra, geometry, measurement, and data analysis. These concepts and skills form the foundation for more advanced mathematical study and are essential for success in many fields and everyday life. The study of elementary mathematics is a crucial part of a student's education and lays the foundation for future academic and career success.
TUTOR, also known as PLATO Author Language, is a programming language developed for use on the PLATO system at the University of Illinois at Urbana-Champaign beginning in roughly 1965. TUTOR was initially designed by Paul Tenczar for use in computer assisted instruction (CAI) and computer managed instruction (CMI) and has many features for that purpose. For example, TUTOR has powerful answer-parsing and answer-judging commands, graphics, and features to simplify handling student records and statistics by instructors. TUTOR's flexibility, in combination with PLATO's computational power, also made it suitable for the creation of games — including flight simulators, war games, dungeon style multiplayer role-playing games, card games, word games, and medical lesson games such as Bugs and Drugs (BND). TUTOR lives on today as the programming language for the Cyber1 PLATO System, which runs most of the source code from 1980s PLATO and has roughly 5000 users as of June 2020.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
Belief perseverance is maintaining a belief despite new information that firmly contradicts it. Such beliefs may even be strengthened when others attempt to present evidence debunking them, a phenomenon known as the backfire effect. For example, in a 2014 article in The Atlantic, journalist Cari Romm describes a study involving vaccination hesitancy. In the study, the subjects expressed their concerns of the side effects of flu shots. After being told that the vaccination was completely safe, they became even less eager to accept them. This new knowledge pushed them to distrust the vaccine even more, reinforcing the idea that they already had before.
