Definition of Algorithm in Mathematics

Definition of Algorithm in Mathematics An algorithm in mathematics is a procedure, a description of a set of steps that can be used to solve a mathematical computation: but they are much more common than that today. Algorithms are used in many branches of science (and everyday life for that matter), but perhaps the most common example is that step-by-step procedure used in long division. The process of resolving a problem in such as what is 73 divided by 3 could be described by the following algorithm: How many times does 3 go into 7?The answer is 2How many are left over? 1Put the 1(ten) in front of the 3.How many times does 3 go into 13?The answer is 4 with a remainder of one.And of course, the answer is 24 with a remainder of 1. The step by step procedure described above is called a long division algorithm. Why Algorithms? While the description above might sound a bit detailed and fussy, algorithms are all about finding efficient ways to do the math. As the anonymous mathematician says, Mathematicians are lazy so they are always looking for shortcuts. Algorithms are for finding those shortcuts. A baseline algorithm for multiplication, for example, might be simply adding the same number over and over again. So, 3,546 times 5 could be described in four steps: How much is 3546 plus 3546? 7092How much is 7092 plus 3546? 10638How much is 10638 plus 3546? 14184How much is 14184 plus 3546? 17730 Five times 3,546 is 17,730. But 3,546 multiplied by 654 would take 653 steps. Who wants to keep adding a number over and over again? There are a set of multiplication algorithms for that; the one you choose would depend on how large your number is. An algorithm is usually the most efficient (not always) way to do the math. Common Algebraic Examples FOIL (First, Outside, Inside, Last) is an algorithm used in algebra that is used in multiplying polynomials: the student remembers to solve a polynomial expression in the correct order: To resolve (4x 6)(x 2), the FOIL algorithm would be: Multiply the first terms in the parenthesis (4x times x 4x2)Multiply the two terms on the outside (4x times 2 8x)Multiply the inside terms (6 times x 6x)Multiply the last terms (6 times 2 12)Add all the results together to get 4x2 14x 12) BEDMAS (Brackets, Exponents, Division, Multiplication, Addition and Subtraction.) is another useful set of steps and is also considered a formula. The BEDMAS method refers to a way to order a set of mathematical operations. Teaching Algorithms Algorithms have an important place in any mathematics curriculum. Age-old strategies involve rote memorization of ancient algorithms; but modern teachers have also begun to develop curriculum over the years to effectively teach the idea of algorithms, that there are multiple ways of resolving complex issues by breaking them into a set of procedural steps. Allowing a child to creatively invent ways of resolving problems is known as developing algorithmic thinking. When teachers watch students do their math, a great question to pose to them is Can you think of a shorter way to do that? Allowing children to create their own methods to resolve issues stretches their thinking and analytical skills. Outside of Math Learning how to operationalize procedures to make them more efficient is an important skill in many fields of endeavor. Computer science continually improves upon arithmetic and algebraic equations to make computers run more efficiently; but so do chefs, who continually improve their processes to make the best recipe for making a lentil soup or a pecan pie. Other examples include online dating, where the user fills out a form about his or her preferences and characteristics, and an algorithm uses those choices to pick a perfect potential mate. Computer video games use algorithms to tell a story: the user makes a decision, and the computer bases the next steps on that decision. GPS systems use algorithms to balance readings from several satellites to identify your exact location and the best route for your SUV. Google uses an algorithm based on your searches to push appropriate advertising in your direction. Some writers today are even calling the 21st century the Age of Algorithms. They are today a way to cope with the massive amounts of data we are generating daily. Sources and Further Reading Curcio, Frances R., and Sydney L. Schwartz. There Are No Algorithms for Teaching Algorithms. Teaching Children Mathematics 5.1 (1998): 26-30. Print.Morley, Arthur. Teaching and Learning Algorithms. For the Learning of Mathematics 2.2 (1981): 50-51. Print.Rainie, Lee, and Janna Anderson. Code-Dependent: Pros and Cons of the Algorithm Age. Internet and Technology. Pew Research Center 2017. Web. Accessed January 27, 2018.

Carbon Family - Element Group 14

Carbon Family - Element Group 14 One way to classify elements is by family. A family consists of homologous element with atoms having the same number of valence electrons and thus similar chemical properties. Examples of element families are the nitrogen family, oxygen family, and carbon family. Key Takeaways: Carbon Family of Elements The carbon family consists of the elements carbon (C), silicon (Si), germanium (Ge), tin (Sn), lead (Pb), and flerovium (Fl).Atoms of elements in this group have four valence electrons.The carbon family is also known as the carbon group, group 14, or the tetrels.Elements in this family are of key importance for semiconductor technology. What Is the Carbon Family? The carbon family is element group 14 of the periodic table. The carbon family consists of five elements: carbon, silicon, germanium, tin,  and lead. It is likely that element 114, flerovium, will also behave in some respects as a member of the family. In other words, the group consists of carbon and the elements directly below it on the periodic table. The carbon family is located very nearly in the middle of the periodic table, with nonmetals to its right and metals to its left. The carbon family is also called the carbon group, group 14, or group IV. At one time, this family was called the tetrels or tetragens because the elements belonged to group IV or as a reference to the four valence electrons of atoms of these elements. The family is also called the crystallogens. Carbon Family Properties Here are some facts about the carbon family: Carbon family elements contain atoms that have 4 electrons in their outer energy level. Two of these electrons are in the s subshell, while 2 are in the p subshell. Only carbon has the s2 outer configuration, which accounts for some of the differences between carbon and other elements in the family.As you move down the periodic table in the carbon family, the atomic radius and ionic radius increase while electronegativity and  ionization energy decrease.  Atom size increases moving down the group because an additional electron shell is added.Element density increases moving down the group.The carbon family consists of one nonmetal (carbon), two metalloids (silicon and germanium), and two metals (tin and lead). In other words, the elements gain metallicity moving down the group.These elements are found in a wide variety of compounds. Carbon is the only element in the group that can be found pure in nature.The carbon family elements have widely variable physical and chemical proper ties.Overall, the carbon family elements are stable and tend to be fairly unreactive. The elements tend to form covalent compounds, though tin and lead also form ionic compounds.Except for lead, all of the carbon family elements exist as different forms or allotropes. Carbon, for example, occurs in diamond, graphite, fullerene, and amorphous carbon allotropes. Tin occurs as white tin, gray tin, and rhombic tin. Lead is only found as a dense blue-gray metal.Group 14 (carbon family) elements have much higher melting points and boiling points than the group 13 elements. Melting and boiling points in the carbon family tend to decrease moving down the group, mainly because atomic forces within the larger molecules are not as strong. Lead, for example, has such a low melting point that its easily liquefied by a flame. This makes it useful as a base for solder. Uses of Carbon Family Elements and Compounds The carbon family elements are important in daily life and in industry. Carbon is the basis for organic life. Its allotrope graphite is used in pencils and rockets. Living organisms, proteins, plastics, food, and organic building materials all contain carbon. Silicones, which are silicon compounds, are used to make lubricants and for vacuum pumps. Silicon is used as its oxide to make glass. Germanium and silicon are important semiconductors. Tin and lead are used in alloys and to make pigments. Carbon Family - Group 14 - Element Facts C Si Ge Sn Pb melting point ( °C) 3500 (diamond) 1410 937.4 231.88 327.502 boiling point ( °C) 4827 2355 2830 2260 1740 density (g/cm3) 3.51 (diamond) 2.33 5.323 7.28 11.343 ionization energy (kJ/mol) 1086 787 762 709 716 atomic radius (pm) 77 118 122 140 175 ionic radius (pm) 260 (C4-) 118 (Sn2+) 119 (Pb2+) usual oxidation number +3, -4 +4 +2, +4 +2, +4 +2, +3 hardness (Mohs) 10 (diamond) 6.5 6.0 1.5 1.5 crystal structure cubic (diamond) cubic cubic tetragonal fcc Source Holt, Rinehart and Winston. Modern Chemistry (South Carolina). Harcourt Education, 2009.