Mathematics: Major Subdisciplines and Concepts

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What are the major subdisciplines of modern mathematics as represented in the text?

Number theory, algebra, geometry, and analysis (correct)

Calculus, statistics, trigonometry, and linear algebra

Differential equations, probability theory, topology, and discrete mathematics

Logic, set theory, combinatorics, and abstract algebra

What does most mathematical activity involve according to the text?

Application of mathematical formulas in practical situations

Experimental testing of mathematical theories

Discovery of properties of abstract objects using pure reason (correct)

Use of mathematical models to represent real-world phenomena

What are the fundamental truths of mathematics independent from according to the text?

Pure reasoning and deductive rules

Natural phenomena

Any scientific experimentation (correct)

Theorems and axioms

What is stipulated to have certain properties in modern mathematics?

Entities Signup and view all the answers

What does a proof in mathematics consist of according to the text?

A succession of applications of deductive rules to already established results Signup and view all the answers

In which academic disciplines is mathematics essential according to the text?

Natural sciences, engineering, medicine, finance, computer science, and social sciences Signup and view all the answers

What is the narrower and more technical meaning of the word 'mathematics' in Classical times?

The study of numbers Signup and view all the answers

Which of the following areas of mathematics had no practical application before its use in the RSA cryptosystem?

Statistics Signup and view all the answers

In Greek mathematics, where did the concept of a proof and its associated mathematical rigour first appear?

Euclid's Elements Signup and view all the answers

What did 'mathēmatikoi' mean in the context of Pythagoreanism?

Learners Signup and view all the answers

In Latin and in English until around 1700, what did the term 'mathematics' more commonly mean?

Astrology Signup and view all the answers

What was one of the two main schools of thought in Pythagoreanism known as?

'Learners' Signup and view all the answers

What was the original meaning of the term 'mathematics' in Latin and English around 1700?

'Astrology' Signup and view all the answers

What was the word 'mathematics' derived from in Ancient Greek?

'Mathematical study' Signup and view all the answers

What did 'mathēmatikós' mean in Ancient Greek?

'Related to learning' Signup and view all the answers

What did 'mathēmatikḗ tékhnē' mean in Ancient Greek?

'The mathematical art' Signup and view all the answers

What does it mean for two figures to be congruent in geometry?

They have the same shape and size. Signup and view all the answers

What is required for two sets of points to be called congruent?

They can be transformed into each other by a combination of rigid motions. Signup and view all the answers

In elementary geometry, when are two line segments considered congruent?

If they have the same length. Signup and view all the answers

What is the condition for two circles to be considered congruent?

They have the same diameter. Signup and view all the answers

When do two plane figures in geometry imply that their corresponding characteristics are 'congruent' or 'equal'?

When their corresponding sides and angles are 'congruent' or 'equal'. Signup and view all the answers

How can congruence of polygons be established graphically?

Match and label the corresponding vertices of the two figures, then draw a vector from one vertex to the corresponding vertex of the other figure, and apply translation, rotation, and reflection. Signup and view all the answers

Which postulate states that if two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent?

ASA (angle-side-angle) Signup and view all the answers

What is the condition for two polygons with n sides to be congruent?

Having numerically identical sequences of sides and angles even if clockwise for one polygon and counterclockwise for the other. Signup and view all the answers

What constitutes sufficient evidence for congruence between two triangles in Euclidean space?

Two pairs of sides equal in length and the included angles equal in measurement. Signup and view all the answers

What do most definitions consider congruence to be?

A form of similarity. Signup and view all the answers