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An algorithm is a detailed set of instructions for solving a problem. One example of an algorithm is the method of long division. Algorithms are named after the mathematician al-Khwarazmi.
Analytic geometry is a branch of mathematics that combines algebra and geometry. Algebraic techniques such as equations are used to understand geometric objects such as lines, curves, and surfaces. Analytic geometry is sometimes called coordinate geometry.
The numerals 1,2,3,4,5,6,7,8,9, and 0 that we use today are often called Arabic numerals because al-Khwarazmi wrote a popular book about them. This number system probably comes from India.
Calculus is a branch of mathematics that involves rates of change, the slopes of curves, length, area, and volume. One important calculus technique is the careful study of very small changes. Sometimes we think of these changes as "infinitely small." Calculus is also concerned with limits and "infinitely large" numbers.
The distance around the outside of a shape is called its circumference. The word "circumference" is usually used for circles; for polygons like squares and triangles, we use the word "perimeter." Sometimes "circumference" is used for three-dimensional objects, like spheres. In this case, the circumference is the distance around the object at its widest point. You might think of this as the distance around the equator of a sphere.
Conic sections are the curves formed when a plane slices through a double cone.
Conic sections include hyperbolas, parabolas, and ellipses. They also include circles, because a circle is a special kind of ellipse. Lines and points can also be made by the intersection of a plane and a double cone, but they usually do not count as conic sections.
The word "convention" has several meanings. The more common meaning is a meeting where people discuss an interest they share. (Conventions of mathematicians can be lots of fun!) But a convention can also be a common way of displaying information. One mathematical convention is using letters from the end of the alphabet, like x, y, and z, to represent variables. Another convention is that numbers on a number line increase from left to right. Conventions do not mean very much on their own, but they make mathematical writing easier to understand.
Coordinates are numbers which are used to locate an object. The word "coordinate" can also be used as an adjective. For example, coordinate geometry is a type of geometry where numbers (coordinates) are used to study lines, curves, and other kinds of shapes.
The coordinate plane is a plane that contains two lines called axes.
Coordinates based on distance from the axes can be used to locate points in the plane.
The diameter of a shape is the greatest possible distance between two points on that shape.
Ellipses are one kind of conic section. They look like ovals, but they have a more precise definition. An ellipse can be described by the equation x2/a2 + y2/b2 = 1.
An ellipse can also be described using foci. An ellipse has two focus points, and the sum of the distances from a point on the ellipse to each focus is always constant. This fact can be written as the equation r1 + r2 = 2a.
A circle is a special kind of ellipse with only one focus.
The equator is the circle around the Earth exactly between the North Pole and the South Pole.
Fermat's Last Theorem
Fermat's Last Theorem states that the equation xn + yn = zn has no solutions when all of the variables are integers (numbers in the set . . . -2, -1, 0, 1, 2, 3, . . .), n is greater than 2, and x, y, and z are not all 0. Pierre de Fermat first stated this theorem, in the margin of one of his books, along with a note that the margin was not big enough to hold the proof. This theorem was not published until after his death. For hundreds of years, nobody could find a correct proof of the theorem. Andrew Wiles finally published a proof in 1995.
A focus is a special point. For example, the center of a circle can be considered its focus. A conic section such as a parabola can be defined by its distance from the focus and a line called the directrix.
The plural of "focus" is "focuses" or "foci."
Hyperbolas are one kind of conic section. A hyperbola can be described by the equation x2/a2 - y2/b2 = 1.
A hyperbola can also be described using foci. A hyperbola has two focus points. The difference of the distances from a point on the ellipse to each focus is always constant. This fact can be written as the equation r2 - r1 = 2a.
The latitude of a point on the Earth is its distance around the Earth from the equator. Since the Earth is a sphere, latitude is measured in degrees. Circles of points on the earth that all have the same latitude are called lines of latitude.
The longitude of a point on the Earth is its distance around the Earth from the Prime Meridian. Since the Earth is a sphere, longitude is measured in degrees. Circles of points on the earth that all have the same longitude are called lines of longitude.
A matrix is made up of numbers arranged in a rectangular shape of rows and columns.
The plural of "matrix" is "matrices." Matrices are important tools in many areas of mathematics, including understanding linear equations.
Optics is the study of light. Optics answers questions such as how mirrors, lenses, and prisms work, and why the sky is blue.
Parabolas are one kind of conic section. A parabola that opens upward can be described by the equation y = ax2 + bx + c.
Two lines in the plane are parallel if they never meet. Parallel lines are always a constant distance from each other.
Plane loci are curves that are defined by their distances from other objects in the plane. The conic sections are plane loci.
In polar coordinates, points are located by their distance from the origin, often labeled r, and their angle from the positive x-axis, often labeled with the Greek letter θ or theta.
The Prime Meridian is a line of longitude that travels from the North Pole to the South Pole through Greenwich, England.
A root of an equation is a place where that equation is zero. For example, y = x2 - 4 has roots at x = 2 and x = -2.
The slope of a line is a number that tells how much it is slanted compared to the x-axis. A line with the equation y = mx + b has slope m. If we draw a tangent line to a curve at a particular point, then we can also define the slope of the curve at that point, by saying that the line and the curve have the same slope at the point where they touch.
A tangent line to a curve only touches the curve at one point.
Technically, the line is allowed to touch the curve again, as long as the next touch is as far away as possible.
There is also a function called the tangent function that is important in trigonometry.
Trigonometry is the study of angles.
Witch of Agnesi
The Witch of Agnesi is a curve. It can be described by the equation y = a3/(x2 + a2).
This curve is named after Maria Gaetana Agnesi, who included it in a calculus textbook she published in 1748. Agnesi called her curve the versiera, which means "turning" in Italian. When her book was translated into English, the translator confused the word "versiera" with the word "avversiera," which means witch, so he called the curve a witch.
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