Calculators

Calculators and Computers

A history of technology that changed the world

General Overview

Calculators are devices that perform mathematical calculations. There are several types of calculators, such as scientific calculators and graphing calculators. However, the first calculators were not electronic, but instead simple tablets of wood. Over the years, advances in technology and the creation of binary in cooperation with circuitry lead to the first electronic calculator/computer. Today, their uses vary, from computing complex operations such as trigonometry and statistical equations to rendering graphical images and graphing functions in 2D and 3D spaces.

Qubits

Bits and Bytes

Qubits, or quantum bits, are essentially bits that that be in the state of 1 and 0 at the same time. To do so, quantum computers such as the Dwave 2 utilize primarily superposition, where particles can exist in different states. Basically, particles are theorized to exist in all possible states at the same time, rather than a particle being in a single state or alternating between several states. However, once a particle in superposition is measured, the property of superposition is lost.

Bits, or Binary digITs, are the 1's and 0's in each place value of binary. Bits are typically not seen alone in binary coding, but instead in 8-bit segments called bytes. With 8 bits in a single byte, 0 to 255 can be represented. Essentially, it is (2^8)-1 because 2^8 has 1 with 8 zeroes behind it, and subtracting one would cause it to now have 8 place values instead of 9.

Byte prefixes:

Using the Babylonian Numbers

The Babylonians had made their own version of their calculator with their numeral system. They could add and subtract very

0 = 00000000

1 = 00000001

2 = 00000010

...

254 = 11111110

255 = 11111111

256 = 100000000

However, bytes are not always 8-bit, as it is broadly defined as a unit of digital information to represent a set group of bits. This definition was created by an American computer scientist, Werner Buchholz, in July 1956. He helped develop IBM's first transistorized supercomputer, the IBM 7030 Stretch. One of the uses of bytes is to hold individual characters in text documents. Each binary value between 0-255 represents a character in the American Standard Code for Information Interchange (ASCII) character set. The phrase "This is binary" saved as a .txt file would have 14 bytes. Each character, including spaces, takes up a byte.

Kilo (K)

2^10 = 1,024

Mega (M)

2^20 = 1,048,576

Giga (G)

2^30 = 1,073,741,824

Tera (T)

2^40 = 1,099,511,627,776

Peta (P)

2^50 = 1,125,899,906,842,624

Exa (E)

2^60 = 1,152,921,504,606,846,976

Zetta (Z)

2^70 = 1,180,591,620,717,411,303,424

Yotta (Y)

2^80 = 1,208,925,819,614,629,174,706,176

Citations

easily and used a variety of formulas to suit different mathematical needs, such as ab = [(a + b)2 - a2 - b2]/2 to help them multiply.

"A Brief History of the Abacus." The Abacus: A Brief History. N.p., n.d. Web. 15 May 2014. <http://www.ee.ryerson.ca/~elf/abacus/history.html>.

"How a calculator works." How a calculator works. N.p., n.d. Web. 16 May 2014. <http://www.vintage-technology.info/pages/calculators/general/calcwork/calcwork.htm>.

"HowStuffWorks "The Base-2 System and the 8-bit Byte"." HowStuffWorks. N.p., n.d. Web. 17 May 2014. <http://computer.howstuffworks.com/bytes1.htm>.

"What is superposition?| Explore | physics.org." What is superposition?| Explore | physics.org. N.p., n.d. Web. 17 May 2014. <http://www.physics.org/article-questions.asp?id=124>.

"quantum art and poetry." : Quantum Computers a classical or objective understanding explaining 'time' as an an emergent property. N.p., n.d. Web. 18 May 2014. <http://quantumartandpoetry.blogspot.com/2013/06/quantum-computers-classical-or.html>.

Hawkley, Catherine. "A Brief Look at the Number Systems and Mathematics of Mesopatamia." : A Brief Look at the Number Systems and Mathematics of Mesopatamia. N.p., n.d. Web. 20 May 2014. <http://reinventingknowledge4.blogspot.com/2011/11/brief-look-at-number-systems-and.html>.

"Calculators: A Pocket-Sized Revolution." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 7. Detroit: Gale, 2009. Global Issues in Context. Web. 23 May 2014.

Document URL

http://go.galegroup.com/ps/i.do?id=GALE%7CTX2643450767&v=2.1&u=tlc049072611&it=r&p=GPS&sw=w&asid=b79271f20fc68ed7f605e34db0656955

Valentine, Nick. "The History of the Calculator." The Calculator Site. N.p., 24 Mar 2014. Web. 24 May 2014. <http://www.thecalculatorsite.com/articles/units/history-of-the-calculator.php>.

Dalakov, Georgi. "History of Computers and Computing, Mechanical calculators, 19th century, Thomas de Colmar." History of Computers and Computing, Mechanical calculators, 19th century, Thomas de Colmar. N.p., n.d. Web. 24 May 2014. <http://history-computer.com/MechanicalCalculators/19thCentury/Colmar.html>.

Fast Forward a Couple Hundred Years...

Where Calculators Started

In 1614, a man named John Napier discovered how to multiply and divide numbers by adding and subtracting them through the use of logarithms {a*b = 10^(log(a)+log(b))}. William Oughtred was the main contributor to the invention of the slide rule in the 17th century using Napier's discovery.

Binary

Adder Circuits

This is a 4 bit circuit that uses the gates mentioned to create the overall circuitry. A1-4 are the first number inputs, while B1-4 are the second. Cin and Cout carry the ones. Modern processors are 64 bit, which would mean there would be 16 of these 4 bit circuits. How all of this

works is, for example, 6+3 is inputted into the calculator. Note

that since the circuit is 4 bit, it is unable to handle numbers larger than 15. The 6 is first translated into binary, which would be 0110, so A1 would be 0, A2 would be 1, and A3 would be 1,and A4 would be 0. The 3 is translated as well, which is 11, with B1&2 being 0 and B3&4 being 1. The signals then go through gates which will emit a 1 or 0, and the final 1s and 0s will be displayed at S1-4. In this case, S1=1, S2=0, S3=0, S4=1. Finally, these signals lead to the display which then converts the binary into decimal.

Binary in Calculators

Subtracting Circuits

Subtracting circuits are actually adder circuits, but with a twist. For example, 6-3 is also 6+(-3). -3 is the "two's compliment" of 3. To find the "two's compliment" of a number, the number is first converted into binary, then the digits are inverted, and finally 1 is added to the inverted digits. The left carry over is ignored. So, 3 is 0011 in binary. The "two's compliment" would be 1101. This is then added to 6, which is 0110. This makes 10011, and since the left carry is ignored, the resulting number is 0011, or 3 in decimal.

Calculators didn't always use binary to solve problems. The Akkadians invented the abacus around 2500 BC for simple calculations, and the Babylonians who lived after them revised it and added a proper written numeral system. Although the abacus was originally invented as a counting tool, people began to use it to aid them in solving various arithmetic problems. The abacus then traveled to China through the Silk Route, who further edited it to suit them. The Babylonian numerals, however, were a calculator in themselves.

Calculators, computers, and other digital electronic circuitry are comprised of gates. The gates allow signals to pass through depending on the the input. Basic gates have two inputs and one output, with each input either being 0 or 1, depending on the voltage. Zero volts results in a zero, 5 volts result in a one. The transistors that make up gates allow the 1 and 0 to take place, as they can either have electricity flowing through or not. The function of these gates are similar to Boolean logic.

Binary, or base 2, is a numerical system where there are only two possible values, 0 and 1. The binary system is the internal programming language of all electronic calculators and computers. This is because transistors within the circuitry can either be in two states, on or off. 1 represents on, 0 represents off. To count in binary, you add 1. If the digit is 1, and you add 1 to it, it becomes a 0 and the 1 carries over to the next value.

This slide rule is set up in a way to multiply numbers by two, as shown by the arrows.

Slide rules began to become more advanced, able to solve square roots and trigonometry problems, but at the same time that the slide rule was invented, mechanic calculators were being looked into.

Three Types of Basic Gates:

AND Gates:

There are two inputs, A and B, and the output Y. The "AND" gate will only give an 1 output if A "AND" B are 1. So if one or both of the inputs are 0, then the output will be 0 as well.

To convert decimal (base 10) to binary (base 2), divide the base 10 number by 2 until you cant continue dividing. As you divide, right down the remainder, either a 1 or 0, right to left.

NAND Gates:

NAND gates are basically "Negative" AND gates, in which the circle notates that the opposite signal is applied. Thus, NAND gates will have the reverse output of AND gates, hence the Negative. If A and/or B is 0, then the output will be 1. The output, Y, will not give a signal if both A and B are 1.

OR Gates:

These gates will give a 1 output if A "OR" B is 1. The output, Y, will not transmit a signal (0) only if both A and B is 0.

Ex: Powers of 2

1= 1 2^1= 2= 10

10= 2 2^2= 4= 100

11= 3 2^3= 8= 1000

100= 4 2^4= 16= 10000

101= 5

110= 6

111= 7

Ex:

123 base 10= x base 2

123/2= 61 r1 → 1

61/2= 30 r1 → 1

30/2= 15 r0 → 0

15/2= 7 r1 → 1 123 base 10= 01111011 base 2

7/2= 3 r1 → 1

3/2= 1 r1 → 1

1/2= 0 r1 → 1

0/2= 0 r0 → 0

The Beginning of the Electronic Calculators

The First Mechanical Calculator

In the 1930's, the world was on the brink of their second war, and the countries wanted to be prepared. One major aspect of the war would be when they dropped bombs, and trigonometry would be necessary to accurately do so. Control systems, such as the Sperry-Norden bombsight, were mechanical calculators integrated with electrical outputs that were connected to weapons. In 1946, the construction of ENIAC (Electronic Numerical Integrator And Computer) marked the first completely digital, artillery firing table calculator. It was 1,000 times faster than the mechanical-electric hybrid calculators and was able to hold a ten digit number in memory, but it weighed 27 tons, 1800 sq. feet, and used as much power as a small town.

The First Electronic Desktop Calculator

In 1961, Control Systems Ltd., located in Britain, invented ANITA (A New Inspiration To Arithmetic/Accounting), the world's first completely electronic calculator for a typical desktop. They had a 12-digit display, and costed around $1,000, or $8,000 in today's currency. Other companies began to invent various other calculating tools, but none were as user friendly as ANITA.

The first hand held calculator was released by Texas Instruments in 1967 called "Cal Tech". It could solve addition, subtraction, multiplication, and division, then print it onto a paper.

Blaise Pascal invented the first mechanical in 1642, but it wasn't fully functional nor ready for mass production. It could add and subtract directly, and multiply and divide by repeated Mechanical calculators were brought back up in 1820 when Thomas de Colmar patented a refined model.