Leibniz`s Harmonic Triangle Paper
... are inverse processes of each other. Using these two ideas, as well as, his concentrated study of Pascal's Triangle, his own Harmonic Triangle, and sums and differences of series, shortly led Leibniz to develop the ideas behind what we now call the Fundamental Theorem of Calculus. Both Pascal's Tria ...
... are inverse processes of each other. Using these two ideas, as well as, his concentrated study of Pascal's Triangle, his own Harmonic Triangle, and sums and differences of series, shortly led Leibniz to develop the ideas behind what we now call the Fundamental Theorem of Calculus. Both Pascal's Tria ...
HSC Mathematics Extension 2
... Representing Complex Numbers on an Argand diagram In previous years the idea of a Number Line was introduced as a continuous line in which real numbers could be placed according to their magnitude and sign. We now must introduce the concept of a Number Plane to accommodate for the Imaginary & Comple ...
... Representing Complex Numbers on an Argand diagram In previous years the idea of a Number Line was introduced as a continuous line in which real numbers could be placed according to their magnitude and sign. We now must introduce the concept of a Number Plane to accommodate for the Imaginary & Comple ...
Algebraic foundations
... The numbers 1, 2, 1 in row 2 are the coefficients of the terms in the expansion of (a + b) 2. (a + b) 2 = 1a2 + 2ab + 1b2 The numbers 1, 3, 3, 1 in row 3 are the coefficients of the terms in the expansion of (a + b) 3. (a + b) 3 = 1a3 + 3a2b + 3ab2 + 1b3 Each row of Pascal’s triangle contains the c ...
... The numbers 1, 2, 1 in row 2 are the coefficients of the terms in the expansion of (a + b) 2. (a + b) 2 = 1a2 + 2ab + 1b2 The numbers 1, 3, 3, 1 in row 3 are the coefficients of the terms in the expansion of (a + b) 3. (a + b) 3 = 1a3 + 3a2b + 3ab2 + 1b3 Each row of Pascal’s triangle contains the c ...
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term ""mathematics"" from the ancient Greek μάθημα (mathema), meaning ""subject of instruction"". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and were transmitted to the west via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.