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.
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 coordinate plane is sometimes called the Cartesian plane in honor of René Descartes.
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 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.
A directrix is a special line. A conic section such as a parabola can be defined by its distance from the directrix and a point called the focus.
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 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.
Nicole Oresme used "latitude" as a more general term for a type of coordinate.
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.
Nicole Oresme used "longitude" as a more general term for a type of coordinate.
A matrix is made up of numbers arranged in a rectangular shape of rows and columns.
Parabolas are one kind of conic section. A parabola that opens upward can be described by the equation y = ax2 + bx + c.
A parabola can also be described using a focus and a directrix. The distance from any point on the parabola to the focus is the same as the distance to the directrix.
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.
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.
There is also a function called the tangent function that is important in trigonometry.
Trigonometry is the study of angles.
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.
