Mathematics is the science of numbers and there are several different branches of mathematical science including algebra, geometry, and calculus. The Merriam-Webster dictionary defines mathematics as the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

Mathematics is not an invention. Discoveries and laws of science are not considered inventions. Inventions are material things and processes. However, there is a history of mathematics, a relationship between mathematics and inventions, and mathematical instruments are considered inventions.

• Mathematics – study of quantity, structure, space, and change. Mathematicians seek out patterns,and formulate new conjectures.

• Arithmetic – oldest and most elementary branch of mathematics, involving the study of quantity, especially as the result of combining numbers. The simplest arithmetical operations include addition, subtraction, multiplication and division.
• Algebra – branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.
• Calculus – branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus is the study of change,^[3] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
• Category theory –
• Discrete mathematics – study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values.
• Combinatorics – branch of mathematics concerning the study of finite or countable discrete structures.
• Geometry – one of the oldest branches of mathematics, it is concerned with questions of shape, size, relative position of figures, and the properties of space.
• Trigonometry – branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

• Other mathematical sciences – academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
  • Statistics – study of the collection, organization, and interpretation of data.It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.
  • Regression analysis – techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.
  • Probability – way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.