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graduates of 2-year programs, and people with less than a 2-year degree or its equivalent in work experience, should face stronger competition for programming jobs. Competition for entry-level positions, however, can even affect applicants with a bachelor's degree. Although demand fluctuates as employer's needs change with technology, prospects should be best for college graduates with knowledge of and experience working with a variety of programming languages and tools, particularly C++ and other object oriented languages such as Smalltalk, Visual Basic, Ada, and Java-as well as newer, domain-specific languages that apply to computer networking, data base management, and Internet applications. In order to remain competitive, college graduates should keep up to date with the latest skills and technologies.

Many employers prefer to hire applicants with previous experience in the field. Employers are increasingly interested in programmers who can combine areas of technical expertise or who are adaptable and able to learn and incorporate new skills. Therefore, individuals who want to become programmers can enhance their chances of doing so by combining the appropriate formal training with practical work experience. Students should try to gain experience by participating in a college work-study program, or undertaking an internship. Students also can greatly improve their employment prospects by taking courses such as accounting, management, engineering, or scienceallied fields in which applications programmers are in demand.

With the expansion of client/server environments, employers will continue to look for programmers with strong technical skills who understand their business and its programming needs. Businesses also look for programmers who develop a technical specialization in areas such as client/server programming, multimedia technology, graphic user interface (GUI), and 4th and 5th generation programming tools. Programmers will be creating and maintaining expert systems and embedding these technologies in more and more products. Other areas of progress include data communications and the business application of Internet technologies. Networking computers so they can communicate with each other is necessary to achieve the greater efficiency organizations require to remain competitive. Demand for programmers with strong object-oriented programming capabilities and experience should arise from the expansion of Intranets, extranets and World Wide Web applications.

Earnings

Median earnings of programmers who worked full time in 1996 were about $40,100 a year. The middle 50 percent earned between about $30,700 and $52,000 a year. The lowest 10 percent earned less than $22,700; the highest 10 percent earned more than $65,200. Starting salary offersfor graduates with a bachelor's degree in the area of computer programming averaged about $35,167 a year in private industry in 1997, according to the National Association of Colleges and Employers. Programmers working in the West and Northeast earned somewhat more than those working in the South and Midwest. On average, systems programmers earn more than applications programmers.

A survey of workplaces in 160 metropolitan areas reported that beginning programmers had median annual earnings of about $27,000 in 1995. Experienced mid-level programmers with some supervisory responsibilities had median annual earnings of about $40,000. Median annual earnings for programmers at the supervisory or team leader level were about $55,000.

According to Robert Half International Inc., starting salaries ranged from $32,500 to $39,000 for programmers and $47,500 to $60,000 for systems programmers in large establishments in 1997. Starting salaries for programmers in small establishments ranged from $28,000 to $37,000.

In the Federal Government, the entrance salary for programmers with a college degree or qualifying experience was about $19,520 a year in early 1997; for those with a superior academic record, $24,180.

Related Occupations Programmers must pay great attention to detail as they write and "debug" programs. Other professional workers who must be detailoriented include computer scientists, computer engineers, and systems analysts, statisticians, mathematicians, engineers, financial analysts, accountants, auditors, actuaries, and operations research analysts.

Sources of Additional Information

State employment service offices can provide information about job openings for computer programmers. Also check with your city's chamber of commerce for information on the area's largest employers. For information about certification as a computing professional,

contact:

Institute for Certification of Computing Professionals (ICCP), 2200 East Devon Ave., Suite 268, Des Plaines, IL 60018. Homepage: http://www.iccp.org Further information about computer careers is available from: The Association for Computing (ACM), 1515 Broadway, New York, NY 10036.

☛ IEEE Computer Society, Headquarters Office, 1730 Massachusetts Ave., NW., Washington, DC 20036-1992.

Computer Scientists, Computer
Engineers, and Systems Analysts

(D.O.T. 030.062-010, .162-014, .167-014; 031; 032; 033; 039; and 109.067-010)

Significant Points

Expected to be the top 3 fastest growing occupations and among the top 20 in the number of new jobs as computer applications continue to expand throughout the economy. A bachelor's degree is virtually a prerequisite for most employers. Relevant work experience also is very important. For some of the more complex jobs, persons with graduate degrees are preferred.

Nature of the Work

The rapid spread of computers has generated a need for highly trained workers to design and develop new hardware and software systems and to incorporate technological advances into new or existing systems. The Handbook refers to this group of professionals as computer scientists, computer engineers, and systems analysts, but in reality this group includes a wide range of professional computer-related occupations. Job titles used to describe this broad category of workers evolve rapidly, reflecting new areas of specialization or changes in technology as well as the preferences and practices of employers. Although many narrow specializations exist, the professional specialty group is commonly referred to as computer scientists, computer engineers, and systems analysts.

The title computer scientist can be applied to a wide range computer professionals who generally design computers and the software that runs them, develop information technologies, and develop and adapt principles for applying computers to new uses. Computer scientists perform many of the same duties as other computer professionals throughout a normal workday, but their jobs are distinguished by the higher level of theoretical expertise and innovation they apply to complex problems and the creation or application of new technology.

Computer scientists can work as theorists, researchers, or inventors. Those employed by academic institutions work in areas ranging from complexity theory, to hardware, to programming language design. Some work on multi-discipline projects, such as developing and advancing uses of virtual reality in robotics. Their counterparts in private industry work in areas such as applying theory, developing specialized languages or information technolo

gies, or designing programming tools, knowledge-based systems, or even computer games.

Computer engineers also work with the hardware and software aspects of systems design and development. Whereas computer scientists emphasize the application of theory, computer engineers emphasize the building of prototypes, although there is much crossover. Computer engineers generally apply the theories and principles of science and mathematics to the design of hardware, software, networks, and processes to solve technical problems. They often work as part of a team that designs new computing devices or computerrelated equipment, systems, or software. Computer hardware engineers generally design, develop, test, and supervise the manufacture of computer hardware-for example, chips or device controllers. Software engineers, on the other hand, are involved in the design and development of software systems for control and automation of manufacturing, business, and management processes. Software engineers or software developers also may design and develop both packaged and systems software or be involved in creating custom software applications for clients. These professionals also possess strong programming skills, but they are more concerned with analyzing and solving programming problems than with simply writing the code for the programs.

Far more numerous, systems analysts use their knowledge and skills to solve computer problems and enable computer technology to meet the individual needs of an organization. They study business, scientific, or engineering data processing problems and design new solutions using computers. This process may include planning and developing new computer systems or devising ways to apply existing systems' resources to additional operations. Systems analysts may design entirely new systems, including both hardware and software, or add a single new software application to harness more of the computer's power. They work to help an organization realize the maximum benefit from its investment in equipment, personnel, and business processes. Most systems analysts generally work with a specific type of system depending on the type of organization they work for-for example, business, accounting or financial systems, or scientific and engineering systems. Companies generally seek business systems analysts who specialize in the type of systems they use.

Analysts begin an assignment by discussing the systems problem with managers and users to determine its exact nature. Much time is devoted to clearly defining the goals of the system and understanding the individual steps used to achieve them so that the problem can be broken down into separate programmable procedures. Analysts then use techniques such as structured analysis, data modeling, information engineering, mathematical model building, sampling, and cost accounting to plan the system. Analysts must specify the inputs to be accessed by the system, design the processing steps, and format the output to meet the users' needs. Once the design has been developed, systems analysts prepare charts and diagrams that describe it in terms that managers and other users can understand. They may prepare cost-benefit and return-on-investment analyses to help management decide whether implementing the proposed system will be financially feasible.

When a system is accepted, analysts determine what computer hardware and software will be needed to set it up. They coordinate tests and observe initial use of the system to ensure it performs as planned. They prepare specifications, work diagrams, and structure charts for computer programmers to follow and then work with them to "debug," or eliminate errors from the system.

In some organizations a single worker called a programmeranalyst is responsible for both systems analysis and programming. (The work of computer programmers is described elsewhere in the Handbook.) As this becomes more commonplace, these analysts will increasingly work with Computer Aided Software Engineering (CASE) tools and object-oriented programming languages, as well as client/server applications development, and multimedia and Internet technology.

One obstacle associated with expanding computer use is the inability of different computer systems to communicate with each other. Because maintaining up-to-date information-accounting records, sales figures, or budget projections, for example-is important in modern organizations, systems analysts may be instructed to make the computer systems in each department compatible so that information can be shared. Many systems analysts are involved with "networking" or connecting all the computers in an individual office, department, or establishment. A primary goal of networking is to allow users to retrieve data from a mainframe computer or a server and use it on their machine. This connection also allows data to be entered into the mainframe from a personal computer. Analysts must design the hardware and software to allow free exchange of data, custom applications, and the computer power to process it all. They study the seemingly incompatible pieces and create ways to link them so users can access information from any part of the system. Networks come in many variations and network systems and data communications analysts design, test, and evaluate systems such as Local Area Networks (LAN), Wide Area Networks (WAN), Internet, and Intranet and other data communications systems. These analysts perform network modeling, analysis and planning, and even research and recommend necessary hardware and software.

Other computer professionals include database administrators and computer support specialists. Database administrators work with database management systems software, coordinating changes to, testing, and implementing computer databases. Since they also may be responsible for design implementation and system security, database administrators plan and coordinate security measures. Computer support specialists provide assistance and advice to users. They interpret problems and provide technical support for hardware, software, and systems. Support specialists may work within an organization or directly for a computer or software vendor. Increasingly, these technical professionals work for help-desk or support services firms, providing customer support on a contract basis to clients as more of this type of work is outsourced.

Many others specialize in analysis, application, or design of a particular system or piece of the system. Network or systems administrators, for example, may install, configure, and support an organizations systems or portion of a system. Telecommunications specialists generally are involved with the interfacing of computer and communications equipment. Computer security specialists are responsible for planning, coordinating, and implementing an organizations' information security measures. These and other growing specialty occupations reflect the increasing emphasis on client-server applications, the growth of the Internet, the expansion of World Wide Web applications and Intranets, and the demand for more end-user support. An example of this is the growing number of job titles relating to the Internet and World Wide Web such as Internet and Web developers, or Webmasters.

Working Conditions

Computer scientists, computer engineers, and systems analysts normally work in offices or laboratories in comfortable surroundings. They usually work about 40 hours a week—the same as many other professional or office workers. However, evening or weekend work may be necessary to meet deadlines or solve specific problems. Given the technology available today, telecommuting is becoming more common for computer professionals. More work, including technical support, can be done from remote locations using modems, laptops, electronic mail, and even through the Internet. It is now possible for technical personnel, such as computer support specialists, to tap into a customer's computer remotely to identify and fix problems.

Like other workers who spend long periods of time in front of a computer terminal typing on a keyboard, computer scientists, engineers, and systems analysts are susceptible to eye strain, back discomfort, and hand and wrist problems such as carpal tunnel syndrome or cumulative trauma disorder.

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Computer scientists, computer engineers, and systems analysts held about 933,000 jobs in 1996, including about 58,000 who were selfemployed. About 216,000 were computer engineers, about 506,000 were computer systems analysts, and about 212,000 were database administrators, computer support specialists, and all other computer scientists.

Although they are employed in most industries, the greatest concentration is in the computer and data processing services industry. This industry includes firms providing nearly every service related to commercial computer use on a contract basis. Services include customized computer programming services and applications and systems software design; the design, development, and production of prepackaged computer software; systems integration, networking, and reengineering services; data processing and preparation services; information retrieval services including on-line data bases and Internet services; on-site computer facilities management; development and management of data bases; and a variety of specialized consulting services. Many others work for government agencies, manufacturers of computer and related electronic equipment, insurance companies, financial institutions, and universities.

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A growing number of computer professionals are employed on a temporary or contract basis-many of whom are self-employed, working independently as contractors or self-employed consultants. For example, a company installing a new computer system may need the services of several systems analysts just to get the system running. Because not all of them would be needed once the system is functioning, the company might contract directly with the systems analysts themselves or with a temporary help agency or consulting firm. Such jobs may last from several months up to 2 years or more. This growing practice enables companies to bring in people with the exact skills they need to complete a particular project, rather than having to spend time or money training or retraining existing workers. Often, experienced consultants then train a company's in-house staff as a project develops.

Training, Other Qualifications, and Advancement

While there is no universally accepted way to prepare for a job as a computer professional because employers' preferences depend on the work to be done, a bachelor's degree is virtually a prerequisite for most employers. Relevant work experience also is very important. For some of the more complex jobs, persons with graduate degrees are preferred.

Computer hardware engineers generally require a bachelor's degree in computer engineering or electrical engineering, whereas software engineers are more likely to need a degree in computer sci

ence. For systems analyst or even database administrator positions, many employers seek applicants who have a bachelor's degree in computer science, information science, computer information systems, or data processing. Computer support specialists may also need a bachelor's degree in a computer-related field, as well as significant experience working with computers, including programming skills. Generally, a Ph.D., or at least a master's degree in computer science or engineering, is required for computer scientist jobs in research laboratories or academic institutions.

Many people develop advanced computer skills in other occupations in which they work extensively with computers, and then transfer into computer occupations. For example, an accountant may become a systems analyst or computer support specialist specializing in accounting systems development, or an individual may move into a systems analyst job after working as a computer programmer.

Regardless of college major, employers generally look for people who are familiar with programming languages and have broad knowledge of and experience with computer systems and technologies, strong problem-solving and analysis skills, and good interpersonal skills. Courses in computer programming or systems design offer good preparation for a job in this field. For jobs in a business environment, employers usually want systems analysts to have a background in business management or a closely related field, while a background in the physical sciences, applied mathematics, or engineering is preferred for work in scientifically oriented organizations. Since employers generally look for experience, entry-level employees enhance their employment opportunities by participating in internship or co-op programs offered through their schools. A related background in the industry in which the job is located, such as financial services, banking, or accounting, can also give an applicant an edge.

Computer scientists, computer engineers, and systems analysts must be able to think logically and have good communication skills. They often deal with a number of tasks simultaneously; the ability to concentrate and pay close attention to detail is important. Although many computer specialists sometimes work independently, they often work in teams on large projects. They must be able to communicate effectively with computer personnel, such as programmers and managers, as well as with users or other staff who may have no technical computer background.

Systems analysts may be promoted to senior or lead systems analysts with experience. Those who show leadership ability also can advance to management positions, such as manager of information systems or chief information officer.

Computer engineers and scientists employed in industry may eventually advance into managerial or project leadership positions. Those employed in academic institutions can become heads of research departments or published authorities in their field. Computer professionals with several years of experience and considerable expertise in a particular subject area or application may find lucrative opportunities as independent consultants or choose to start their own computer consulting firms.

Technological advances come so rapidly in the computer field that continuous study is necessary to keep skills up to date. Continuing education is usually offered by employers, hardware and software vendors, colleges and universities, or private training institutions. Additional training may come from professional development seminars offered by professional computing societies.

As technology becomes more sophisticated and complex, a higher level of skill and expertise is demanded by employers in all areas. Technical or professional certification is becoming a more common way to ensure employers of a level of competency or quality in a prospective employee. Many product vendors offer and may even require professionals who work with their products to be certified. Voluntary certification is also available through organizations such as the Institute for Certification of Computing Professionals (ICCP). ICCP offers the designation Certified Computing Professional (CCP) to those who have at least 4 years of work experience as a computer professional, or at least 2 years experience and a college degree. Candidates must pass a

core examination testing general knowledge, plus exams in two specialty areas, or in one specialty area and two computer programming languages. The Quality Assurance Institute (QAI) awards the designation Certified Quality Analyst (CQA) to those who meet education and experience requirements, pass an exam, and endorse a code of ethics. Neither designation is mandatory, but professional certification may provide a job seeker a competitive advantage.

Job Outlook

Computer scientists, computer engineers, and systems analysts are expected to be the three fastest growing occupations through the year 2006. Employment of computing professionals is expected to increase much faster than average as technology becomes more sophisticated and organizations continue to adopt and integrate these technologies, making for plentiful job openings. Growth will be driven by very rapid growth in computer and data processing services, which is projected to be the fastest growing industry. addition, thousands of job openings will result annually from the need to replace workers who move into managerial positions or other occupations or who leave the labor force.

In

Computer scientists, computer engineers, and systems analysts will need to continually upgrade their technical expertise and improve their ability to interact with users as the sophistication and complexity of technology advances. As more computing power is made available to the individual user and users develop more sophisticated knowledge of computers, they become more aware of the machine's potential and better able to suggest how computers could be used to increase their own productivity and that of the organization. Increasingly, users are able to design and implement more of their own applications and programs. The result is a growing demand for computer support specialists, help desk personnel, and technical consultants.

The demand for "networking" to facilitate the sharing of information, the expansion of client/server environments, and the need for specialists to use their knowledge and skills in a problem solving capacity will be a major factor in the rising demand for systems analysts. Falling prices of computer hardware and software should continue to induce more businesses to expand computerized operations and integrate new technologies. In order to maintain a competitive edge and operate more cost effectively, firms will continue to demand computer professionals who are knowledgeable about the latest technologies and able to apply them to meet the needs of businesses.

New growth areas generally arise from the development of new technologies. Therefore, it is important for computer professionals at all levels to keep their skills up to date. The expanding integration of Internet technologies by businesses, for example, has resulted in a rising demand for a variety of skilled professionals who can develop and support Internet, Intranet, and World Wide Web applications. Growth in these areas is also expected to create demand for computer scientists, computer engineers, and systems analysts knowledgeable about network, data and communications security.

Since employers look for the most qualified applicants possessing a high level of technical expertise, individuals with an advanced degree in computer science, management information systems (MIS), computer engineering, or an MBA with a concentration in information systems should enjoy very favorable employment prospects. College graduates with a bachelor's degree in computer science, computer engineering, information science, or information systems should also enjoy very favorable prospects, particularly if they have supplemented their formal education with some level of practical experience. College graduates with non-computer science majors who have had courses in computer programming, systems analysis, and other data processing areas, as well as training or experience in an applied field, should also be able to find jobs as computer professionals. Those who are familiar with client/server environments, CASE tools and object-oriented programming, Internet, Intranet, and multimedia technology will have an even greater advantage, as will individuals with significant networking, database, and systems experience. Employers will continue to seek computer professionals who can combine strong programming and

traditional systems analysis skills with good interpersonal and business skills.

Earnings

Median annual earnings of computer systems analysts and scientists who worked full time in 1996 were about $46,300. The middle 50 percent earned between $34,000 and $59,900. The lowest 10 percent earned less than $24,800 and the highest tenth, more than $76,200. Computer scientists with advanced degrees generally earn more than systems analysts.

Starting salaries for computer scientists or computer engineers with a bachelor's degree can be significantly higher than starting salaries of bachelor's degree graduates in many other fields. According to the National Association of Colleges and Employers, starting salary offers for graduates with a bachelor's degree in computer engineering averaged about $39,722 a year in 1997; those with a master's degree, $44,734 a year; and those with a Ph.D., $63,367. Starting offers for graduates with a bachelor's degree in computer science averaged about $36,597 a year; in information sciences, about $35,407 a year; and in systems analysis, about $43,800 a year in 1997. Offers for those with the bachelor's degree vary by functional area for all types of employers, as shown in the following tabulation.

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Offers for graduates with a master's degree in computer science in 1997 averaged $45,853 a year; and those with a Ph.D. in computer and information sciences, $61,306.

According to Robert Half International Inc., starting salaries in 1997 for systems analysts employed by large establishments employing more than 50 staff members ranged from $46,000 to $57,500. Salaries for those employed in small establishments ranged from $38,000 to $48,000. Salaries for programmer-analysts ranged from $39,000 to $50,000 in large establishments and $33,500 to $43,000 in small establishments. Starting salaries ranged from $54,000 to $67,500 for data base administrators, from $36,000 to $55,000 for network administrators, from $25,000 to $36,500 for help desk support technicians, and from $49,000 to $67,500 for software development specialists.

In the Federal Government, the entrance salary for systems analysts who are recent college graduates with a bachelor's degree was about $19,520 a year in early 1997; for those with a superior academic record, $24,180. The average annual salary for computer engineers in the Federal Government in nonsupervisory, supervisory, and managerial positions was $62,900 in early 1997.

Related Occupations

Other workers who use research, logic, and creativity to solve business problems are computer programmers, financial analysts, urban planners, engineers, mathematicians, statisticians, operations research analysts, management analysts, and actuaries.

Sources of Additional Information

Further information about computer careers is available from:

Association for Computing (ACM), 1515 Broadway, New York, NY 10036. ☛ IEEE Computer Society, Headquarters Office, 1730 Massachusetts Ave., NW., Washington, DC 20036-1992.

Information about the designation Certified Computing Professional is available from:

Institute for Certification of Computing Professionals (ICCP), 2200 East Devon Ave., Suite 268, Des Plaines, IL 60018. Homepage: http://www.iccp.org Information about the designation Certified Quality Analyst is available from:

Quality Assurance Institute, 7575 Dr. Phillips Blvd., Suite 350, Orlando, FL 32819.

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Mathematicians

(D.O.T. 020.067-014, .167-030; 199.267-014)

Significant Points

Employment is expected to grow more slowly than the average as civilian and defense related research activities face limited expansion.

Bachelor's degree holders with a strong background in

computer science, electrical or mechanical engineering, or operations research should have good opportunities in related occupations.

Nature of the Work

Mathematics is one of the oldest and most fundamental sciences. Mathematicians create new mathematical theories and techniques involving the latest technology and solve economic, scientific, engineering, and business problems using mathematical knowledge and computational tools.

Mathematics falls into two broad classes: theoretical (pure) mathematics and applied mathematics. However, these classes are not sharply defined and often overlap.

Theoretical mathematicians advance mathematical knowledge by developing new principles and recognizing previously unknown relationships between existing principles of mathematics. Although they seek to increase basic knowledge without necessarily considering its practical use, such pure and abstract knowledge has been instrumental in producing or furthering many scientific and engineering achievements.

Applied mathematicians use theories and techniques, such as mathematical modeling and computational methods, to formulate and solve practical problems in business, government, engineering, and the physical, life, and social sciences. For example, they may analyze the most efficient way to schedule airline routes between cities, the effects of new drugs on disease, the aerodynamic characteristics of an experimental aircraft, or the distribution costs or manufacturing processes of businesses. Applied mathematicians working in industrial research development may develop or enhance mathematical methods when confronted with difficult problems. Some mathematicians, called cryptanalysts, analyze and decipher encryption systems designed to transmit military, political, financial, or law enforcement-related information.

and

Mathematicians use computers extensively to analyze relationships among variables, solve complex problems, develop models, and process large amounts of data.

Much work in applied mathematics, however, is carried on by persons with titles other than mathematician. In fact, because mathematics is the foundation upon which so many other academic disciplines are built, the number of workers using mathematical techniques is many times greater than the number actually designated as mathematicians.

Engineers, computer scientists, physicists, and economists are among those who use mathematics extensively but have job titles other than mathematician. Some workers, such as statisticians, actuaries, and operations research analysts, actually are specialists in a particular branch of mathematics. (See statements on actuaries, operations research analysts, and statisticians elsewhere in the Handbook.)

Working Conditions

Mathematicians working for government agencies or private firms usually have structured work schedules. They generally work as an integral part of a interdisciplinary team that may include engineers, computer scientists, physicists, technicians, and others. Deadlines, overtime work, special rush requests for information or analysis, and prolonged travel to attend seminars or conferences may be part of their jobs.

Mathematicians generally work as an integral part of a interdisciplinary team.

Employment

Mathematicians held about 16,000 jobs in 1996. In addition, about 20,000 persons held mathematics faculty positions in colleges and universities in 1995, according to the American Mathematical Society. (See the statement on college and university faculty elsewhere in the Handbook.)

Many nonfaculty mathematicians work for either Federal or State governments. The Department of Defense is the primary Federal employer of mathematicians, employing almost three-fourths of the mathematicians employed by the Federal Government. In the private sector, major employers include research and testing services, educational services, security and commodity exchanges, and management and public relations services. Within manufacturing, the drug industry is the key employer. Some mathematicians also work for banks, insurance companies, and public utilities.

Training, Other Qualifications, and Advancement

A bachelor's degree in mathematics is the minimum education needed for prospective mathematicians. In the Federal Government, entry-level job candidates usually must have a 4-year degree with a major in mathematics or a 4-year degree with the equivalent of a mathematics major-24 semester hours of mathematics

courses.

In private industry, job candidates generally need a master's or a Ph.D. degree to obtain jobs as mathematicians. Most of the positions designated for mathematicians are in research and development laboratories as part of technical teams. These research scientists engage in either pure mathematical, or basic, research; or in applied research focusing on developing or improving specific products or processes. The majority of those with a bachelor's or master's degree in mathematics who work in private industry do so not as mathematicians, but in related fields such as computer science, where they have titles such as computer programmers, systems analysts, or systems engineers.

A bachelor's degree in mathematics is offered by most colleges and universities. Mathematics courses usually required for this degree are calculus, differential equations, and linear and abstract algebra. Additional courses might include probability theory and statistics, mathematical analysis, numerical analysis, topology, discrete mathematics, and mathematical logic. Many colleges and universities urge or even require students majoring in mathematics

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