{"id":4460,"date":"2016-10-03T21:24:17","date_gmt":"2016-10-03T21:24:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&p=4460"},"modified":"2018-05-17T00:00:39","modified_gmt":"2018-05-17T00:00:39","slug":"introduction-12","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/introduction-12\/","title":{"raw":"Why learn about roots and rational exponents?","rendered":"Why learn about roots and rational exponents?"},"content":{"raw":"[caption id=\"attachment_5424\" align=\"alignright\" width=\"293\"]\"Circle, Circle, ellipse, parabola and hyperbola.[\/caption]\r\n\r\nOne of the most well known\u00a0uses of roots and rational exponents is found in quadratic equations. Quadratic comes from quadratus,<\/i>\u00a0the Latin word\u00a0<\/i>for square.\u00a0When we factor a quadratic equation, we are completing the square.\r\n\r\nHumans have been working with\u00a0quadratic equations for over two\u00a0thousand years\u00a0in many areas around the world, including India,\u00a0Egypt, China, and Greece. The circle, ellipse, hyperbola, and parabola are all examples of quadratic curves, which have been studied since the ancient Greeks. Of these, however, only the circle was relevant to the real world until the\u00a0beginning of the Renaissance in the 16th century.\r\n\r\nThe Renaissance period was a rediscovery of Greek philosophy which, as you probably realize by now, includes a significant amount of mathematical principles. In addition to the rediscovery of old ideas, Renaissance thinkers were discovering new facts about the world around them, some of which challenged the ideas of the ancient Greeks.\r\n\r\nFor example, in 1543 Copernicus published his theory that the Earth rotated\u00a0around the Sun, which was the exact opposite of what most people in Western culture believed at the time. Copernicus thought that the orbit of the Earth was a circle, which was regarded as the most perfect possible curve because of its symmetry.\r\n\r\nIn 1609 Kepler published his first two laws of planetary motion. Building\u00a0off of Tycho Brahe's work, Kepler discovered that instead of circles\u00a0planets move around the sun\u00a0in ellipses.<\/i>\u00a0Other objects, Kepler found, may also follow a parabola\u00a0<\/em>or\u00a0hyberbola<\/em>\u00a0which, along with ellipses, belong to a group of curves known as conic sections.\r\n\r\nThe invention of the telescope in the 17th century allowed us to learn even more about the planets. Using a telescope he created,\u00a0Galileo was able to observe\u00a0the moon,\u00a0discover Jupiter's four satellites, verify the phases of Venus, and prove the validity of Copernicus' theory of heliocentrism.\u00a0His\u00a0telescope used lenses, the shape of which was formed by two intersecting hyperbolae. Newton's reflecting telescope includes\u00a0a mirror for which each cross section takes the shape of a parabola. Even the Hubble Space Telescope uses paraboloidal mirrors. As unbelievable as it may seem, our world of modern communications, from Facebook to GPS to planetary exploration, is made possible thanks to quadratic equations!","rendered":"
\"Circle,<\/p>\n

Circle, ellipse, parabola and hyperbola.<\/p>\n<\/div>\n

One of the most well known\u00a0uses of roots and rational exponents is found in quadratic equations. Quadratic comes from quadratus,<\/i>\u00a0the Latin word\u00a0<\/i>for square.\u00a0When we factor a quadratic equation, we are completing the square.<\/p>\n

Humans have been working with\u00a0quadratic equations for over two\u00a0thousand years\u00a0in many areas around the world, including India,\u00a0Egypt, China, and Greece. The circle, ellipse, hyperbola, and parabola are all examples of quadratic curves, which have been studied since the ancient Greeks. Of these, however, only the circle was relevant to the real world until the\u00a0beginning of the Renaissance in the 16th century.<\/p>\n

The Renaissance period was a rediscovery of Greek philosophy which, as you probably realize by now, includes a significant amount of mathematical principles. In addition to the rediscovery of old ideas, Renaissance thinkers were discovering new facts about the world around them, some of which challenged the ideas of the ancient Greeks.<\/p>\n

For example, in 1543 Copernicus published his theory that the Earth rotated\u00a0around the Sun, which was the exact opposite of what most people in Western culture believed at the time. Copernicus thought that the orbit of the Earth was a circle, which was regarded as the most perfect possible curve because of its symmetry.<\/p>\n

In 1609 Kepler published his first two laws of planetary motion. Building\u00a0off of Tycho Brahe’s work, Kepler discovered that instead of circles\u00a0planets move around the sun\u00a0in ellipses.<\/i>\u00a0Other objects, Kepler found, may also follow a parabola\u00a0<\/em>or\u00a0hyberbola<\/em>\u00a0which, along with ellipses, belong to a group of curves known as conic sections.<\/p>\n

The invention of the telescope in the 17th century allowed us to learn even more about the planets. Using a telescope he created,\u00a0Galileo was able to observe\u00a0the moon,\u00a0discover Jupiter’s four satellites, verify the phases of Venus, and prove the validity of Copernicus’ theory of heliocentrism.\u00a0His\u00a0telescope used lenses, the shape of which was formed by two intersecting hyperbolae. Newton’s reflecting telescope includes\u00a0a mirror for which each cross section takes the shape of a parabola. Even the Hubble Space Telescope uses paraboloidal mirrors. As unbelievable as it may seem, our world of modern communications, from Facebook to GPS to planetary exploration, is made possible thanks to quadratic equations!<\/p>\n","protected":false},"author":21,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4460","chapter","type-chapter","status-publish","hentry"],"part":774,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4460\/revisions"}],"predecessor-version":[{"id":5267,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4460\/revisions\/5267"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/774"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4460\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=4460"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4460"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=4460"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=4460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}