Mathematics in the Modern World Lecture Flashcards

Mathematics

A field of study that explores numbers, quantities, shapes, structures, and patterns.

Archimedes

A Greek mathematician considered by many as the "Father of mathematics" because of his notable inventions in mathematics and science.

Carl Friedrich Gauss

A German mathematician and physicist referred to as the "Prince of Mathematics," famous for discoveries in number theory, electromagnetism, and matrices.

Galileo Galilei

An Italian astronomer, philosopher, physicist, and mathematician described as the "father of observational astronomy" and "father of modern science and physics" who made revolutionary telescopic discoveries.

Albert Einstein

A physicist (1879-1955) famous for his mass and energy equation, $E = mc^2$, whose theory of relativity relied heavily on mathematics.

Isaac Newton

The scientist who laid down natural laws such as the law of freely falling bodies, and co-developed modern calculus in the 17th century.

Gottfried Leibniz

The mathematician who, alongside Isaac Newton, developed modern calculus in the 17th century.

Cartesian Coordinate System

The fusion of Geometry and Algebra created by Rene Descartes (1596-1650).

Fibonacci Sequence

A series of numbers in which each number is the sum of the two preceding numbers, commonly denoted as $F_n$, starting as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.

Bilateral Symmetry

A pattern found in nature for which the human body is an excellent example of a living being.

Ocean Gyre

A large system of circular ocean currents formed by global wind patterns and forces created by Earth's rotation.

Leonardo Fibonacci

An Italian mathematician born in Pisa, Italy, around 1175 AD (also known as Leonardo Pisano or Leonard of Pisa), who introduced the concept of zero and the Hindu-Arabic numeral system to Europe.

Liber Abaci

A famous book written by Fibonacci in 1202 that introduced the Hindu-Arabic number system to Europe and posed a population growth problem involving rabbits.

Binet's Formula

A formula derived by Jacques Binet in 1843 used to find the $n\text{th}$ Fibonacci Number.

Golden Ratio

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$\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, 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$\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… no, $\frac{1+\frac{\text{1}}{\text{1}}}{\text{2}}$… wait, defined as $$rac{1+rac{ ext{1}}{ ext{1}}}{ ext{2