Personal computers

A computer is a device that manipulates raw data into potentially useful information. Computers may be analog or electronic. Analog computers use mechanical elements to perform functions. For example, Stonehenge in England is believed by some to be an analog computer. It allegedly uses the stones along with the positions of the sun and moon to predict celestial events like the solstices and eclipses. Electronic computers use electrical components like transistors for computations.

Many consider the first personal computer to be Sphere 1, created by Michael Wise in the mid-1970s. The Apple II was introduced in 1977, and Apple Inc. offered the Macintosh, which had the first mass-marketed graphical user interface, by 1984. IBM debuted its personal computer in 1981. “Macs” and PCs quickly became common in businesses and schools for a variety of purposes. Processing speed, size, memory capacity, and other functional components have become faster, smaller, lighter, and cheaper over time, and personal computers have evolved into a multitude of forms designed to be customizable to each user’s needs. At the beginning of the twenty-first century, desktops, laptops, netbooks, tablet PCs, palm-sized smartphones, handheld programmable calculators, digital book readers, and devices like Apple’s iPad offer access to computing, the Internet, and other functions.

Mathematical History of Computers

Modern computing can be traced to nineteenth century mathematician Charles Babbage’s analytical engine. Boolean algebra, devised by mathematician George Boole later in the same century, provided a logical basis for digital electronics. Lambda calculus, developed by mathematician Alonso Church in the early twentieth century, also laid the foundations for computer science, while the Turing machine, a theoretical representation of computing developed by mathematician Alan Turing, essentially modeled computers before they could be built. In the 1940s, mathematicians Norbert Wiener and Claude Shannon researched information control theory, further advancing the design of digital circuits. The Electrical Numerical Integrator and Calculator (ENIAC) was the first general purpose electronic computer. It was created shortly after World War II by physicist-engineer John Mauchly and engineer J. Presper Eckert. They also developed the Binary Automatic Computer (BINAC), the first dual-processor computer, which stored information on magnetic tape rather than punch cards, and the first commercial computer, Universal Automatic Computer (UNIVAC). Mathematician John Von Neumann made important modifications to ENIAC, including serial operations to facilitate mathematical calculations. Scientists William Bradford Shockley, John Bardeen, and Walter Brattain won the 1956 Nobel Prize in Physics for transistor and semiconductor research, which influenced the development of most subsequent electronic devices, including personal computers. During the latter half of the twentieth century, countless mathematicians, computer scientists, engineers, and others advanced the science and technology of personal computers, and research has continued into the twenty-first century. For example, Microsoft co-founder Bill Gates published a paper on sorting pancakes, which has extensions in the area of computer algorithms. Personal computers have facilitated mathematics teaching and research in many areas such as simulation, visualization, and random number generation, though the use of calculators and software like Maple for teaching mathematics generated controversy.

Devices, Memory, and Processor Speeds

The typical personal computer has devices for the input and output of information and a means of retaining programs and data in memory. It also has the means of interacting with programs, data, memory, and devices attached to the computer’s central processing unit (CPU). Input devices have historically included a keyboard and a mouse, while newer systems frequently use touch technology, either in the form of a special pad or directly on the screen. Other devices include scanners, digital cameras, and digital recorders. Memory storage devices are classified as “primary memory” or “secondary” devices. The primary memory consists of the chips on the board inside the case of the computer. Primary memory comes in two types: read only memory (ROM) and random access memory (RAM). ROM contains the rudimentary part of the operating system, which controls the interaction of the computer components. RAM holds the programs and data while the computer is in use. The most popular types of secondary memory used for desktop computers include magnetic disk drives, optical CD and DVD drives, and USB flash memory.

The speed of the computer operation is an important factor. Computers use a set clock cycle to send the voltage pulses throughout the computer from one component to another. Faster processing enables computers to run larger, more complex programs. The disadvantage is that heat builds up around the processor, caused by electrical resistance. ENIAC was 1000 times faster than the electromechanical computers that preceded it because it relied on vacuum tubes rather than physical switches. Turing made predictions regarding computer speeds in the 1950s, while Moore’s law, named for Intel co-founder Gordon Moore, quantified the doubling rate for transistors per square inch on integrated circuits. The number doubled every year from 1958 into the 1960s, according to Moore’s data. The rate slowed through the end of the twentieth century to roughly a doubling every 18 months. Some scientists predict more slowdowns because of the heat problem. Others, like mathematician Vernor Vinge, have asserted that exponential technology growth will produce a singularity, or essentially instantaneous progress. Processing speed, memory capacity, pixels in digital images, and other computer capabilities have been limited by this effect. There has also been a disparity in the growth rates of processor speed and memory capacity, known as memory latency, which has been addressed in part by mathematical programming techniques, like caching and dynamic optimization.

Carbon nanotubes and magnetic tunnels might be used to produce memory chips that retain data even when a computer is powered down. At the start of the twenty-first century, this approach was being developed with extensive mathematical modeling and physical testing. Other proposed solutions involved biological, optical, or quantum technology. Much of the physics needed for quantum computers exists only in theory, but mathematicians like Peter Shor began working on the mathematics of quantum programming in the early twenty-first century, which involves ideas like Fourier transforms, periodic sequences, prime numbers, and factorization. Fourier transforms are named for mathematician Joseph Fourier.

The Digital Divide

The digital divide is the technology gap between groups that have differential access to personal computers and related technology. The gap is measured both in social metrics, such as soft skills required to participate in online communities, and infrastructure metrics, such as ownership of digital devices. Mathematical methods are used to quantify the digital divide. Comparisons may be made using probability distributions and Lorenz curves, developed by economist Max Lorenz, and measures of dispersion such as the Gini coefficient, developed by statistician Corrado Gini. Researchers have found digital divides among different countries, and within countries, among people of different ages, between genders, and among socioeconomic strata.

The global digital divide quantifies the digital divides among countries and is typically given as the differences among the average numbers of computers per 100 citizens. In the early twenty-first century, this metric varied widely. Several concerted private and government efforts, such as One Laptop Per Child, were directed at reducing the global digital divide by providing computers to poor countries. The breakthroughs connected to these efforts, such as mesh Internet access architecture, benefited all users. The Digital Opportunity Index (DOI) is computed by the United Nations based on 11 metrics of information and communication technologies, such as proportion of households with access to the Internet. It has been found to be positively associated with a country’s wealth.

Bibliography

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"Personal Computers." Computer History Museum, www.computerhistory.org/brochures/personal-computers/. Accessed 20 Nov. 2024.

"Personal Computer (PC)." Britannica, 19 Nov. 2024, www.britannica.com/technology/personal-computer. Accessed 20 Nov. 2024.

Schneider, Josh, and Ian Smalley. "What Is Quantum Computing?" IBM, 5 Aug. 2024, www.ibm.com/topics/quantum-computing. Accessed 20 Nov. 2024.

Wozniak, Steve, and Gina Smith. iWoz: Computer Geek to Cult Icon: How I Invented the Personal Computer, Co-Founded Apple, and Had Fun Doing It. W. W. Norton, 2007.