The Pythagorean theorem, the Fundamental Theorem of Calculus, and the Euler characteristic are all examples of mathematical connections between different areas of mathematics.

Infinity is the limit of a sequence that approaches a certain value, limits are used to define infinity, and infinity and limits are both used to describe the behavior of functions.

Symmetry groups are used to classify different types of symmetries, group theory is used to study the properties of symmetry groups, and symmetry and group theory are both used to study the structure of mathematical objects.

The use of mathematics in physics to model the motion of objects, the use of mathematics in biology to model the growth of populations, and the use of mathematics in economics to model the behavior of markets are all examples of mathematical connections between mathematics and other fields.

Probability is used to calculate the likelihood of an event occurring, statistics is used to collect and analyze data, and probability and statistics are both used to make predictions about the future.

The use of complex numbers in quantum mechanics, the use of group theory in chemistry, and the use of topology in computer science are all examples of mathematical connections between two different areas of mathematics.

Calculus is used to study the rate of change of functions, analysis is used to study the properties of functions, and calculus and analysis are both used to study the behavior of functions.

Algebraic equations can be used to represent geometric shapes, geometric figures can be used to illustrate algebraic concepts, and algebra and geometry are both used to study the structure of mathematical objects.

The use of mathematics in engineering to design bridges and buildings, the use of mathematics in medicine to develop new drugs and treatments, and the use of mathematics in finance to model the behavior of stock markets are all examples of mathematical connections between mathematics and another field.

Number theory is used to develop cryptographic algorithms, cryptography is used to protect data from unauthorized access, and number theory and cryptography are both used to study the properties of numbers.

The use of linear algebra in computer graphics, the use of topology in knot theory, and the use of category theory in algebraic geometry are all examples of mathematical connections between two different areas of mathematics.

Logic is used to study the properties of sets, set theory is used to develop logical systems, and logic and set theory are both used to study the foundations of mathematics.

Calculus is used to study the motion of objects, physics is used to study the forces that act on objects, and calculus and physics are both used to study the behavior of the universe.

The use of mathematics in economics to model the behavior of markets, the use of mathematics in psychology to study human behavior, and the use of mathematics in sociology to study social networks are all examples of mathematical connections between mathematics and another field.

Probability is used to study the likelihood of events occurring, combinatorics is used to study the number of ways that objects can be arranged, and probability and combinatorics are both used to study the behavior of random variables.