Calculators

Simple Caculators

Mathematics could be calculated on paper but this was somewhat error prone. Most of us have done some math problems and made a mistake a time or two.

People understood this, but they also saw that many of the calculations they were doing were repeatable and so they started making tools that helped with these.

At first people just used beads to keep track of their current values as they added them up. This meant they no longer had to write ever calculation on paper or keep it in their brain. Abacus strung these beads onto poles keeping it in a rigid frame and could be used to keep track of addition subtraction, division, and multiplication of large numbers.

People still had to do small math sequences but it made it easiser.

Put althought this eliminated some paper and error, people were still repeating the same calculations many times over, so it could be improved. Even when doing large calculations it was just composed of many of the same smaller calculations

A brilliant mathematition devloped a manual way that you could move tiles around that broke large calculations into small pieces. This was called Napier's Bones and allowed people to speed up calculation even further by already seeing the small calculations needed in front of them.

People kept inovatting thorugh. By using the principles of geared teeth it was found that you could "spin" gears and rotate display drums that would represent the numbers. This allowed full automated calculations where you just had to move the gears to put in your starting number and by the gears moving it would create the correct output.

In 1642 Blaise Pascal created one of the first advance gear based calculators called the pascaline which was originally used to calculate taxes.

And in 1673 the Stepped Reckoner, created by Gottfried Wilhelm Leibniz, expanded on this allowing for 4 operations (addition, subtraction, multiplication, and division) to be included. It improved upon gears using a stepped drum. This design would be standardized as the most common mechanical calculator for over 200 years.

Electronic Calculators

Eventually mathematics showed that we could do math using boolean algebra, and people saw that each step of the boolean algebra was similar to that of a switch or relay. Putting these two technologies together they saw that you could make a calculator using just relays. By setting the initial relays on 2 inputs sides the corerct wiring would lead to a single solution on the output side, making a calculator. They improved this by using punch cards instead of having to manually set the switches, and thus could rapidly do calculations. This intial relay based electric calculator shows that it was possible but it took thousands of relays and was the size of a room in a house.

Eventually it was shown that electronics could be used to make calculations. Instead of complex gears which had many points of contact and abliity to get jammed. It was shown that Relays(LINK: to electronic history section) and binary math (LINK to math history section) could be used to make calculations.

Although these first iterations were incredibly large. The input and output could be controlled much quicker and the calculation was almost instant as relays could switch much faster than gears could turn.

The key insight was that relays could be wired together to create "logic gates" - fundamental building blocks that perform boolean operations. By combining these gates, any mathematical calculation could be performed electronically.

These logic gates formed the foundation of all digital computing. By combining thousands of these simple gates, early engineers built machines capable of performing complex arithmetic automatically.

Below is a demonstration of how a relay-based computer performs addition. Each logic gate is represented as a block with a relay bar indicator showing its state. Enter two numbers and watch as the binary signals propagate through the circuit.

This ripple carry adder demonstrates the fundamental principle: by chaining together half adders and full adders built from logic gates, we can add arbitrarily large binary numbers. Each bit position processes the inputs and generates both a sum bit and a carry bit for the next position.

Transition to the computer

Now that a computer was quick but large, making it smaller was the next step.

The invention of vaccum tubes and then transistors was the next step.

Instead of relays which required a moving metal plate to control electriicty flow, these new technologies were able to use physics properties to control the flow of electricity entirely without mechanical movement.

These were able to do this just using physcial properties of materials. Allowing to allow or stop electiricty flow entirely using applying electical voltage to differnt locations upon what is called a transistor

Transistors made computers even quicker than before. More durable through their "solid" components, and smaller and more energy efficient using these new materials.

Computers Computing Algorithms

These computers could now use all of the algorithsm that were devleoped for boolean math and could now reprsent the input and output of any mathemtacial sequence.

Incorporating Data Inputs

For repeatable calculations the inputs could be programmed using punch cards. These were early forms of data storage and retrieval. Attributes could be programmed/punched into cards and then used and or reused.

This increased the speed of setup and data input for alogrithms.

Eventually this technology further improve and got smaller and more efficient by using metal embedded in strips of acetate plastic. This moved the size of a punch from milimeters to micrometers instead.

From here on everything kept getting more compact and efficient. Instead of individual transistors being wired together they were made into singular "chips". The desnity of these grew exponentially from hundreds to thousands, hundreds of thousands, and millions and now billions

Storage went from magnetic strips, to magnetic drums, to magnetic disks (HDDs Hard Disk Drives) to using speicalized transitors to capture data to make it even more efficient just as was done for procesors.

The input and output methods chagned aswell. data was read and displayed onto screens represeting numbers and letters. Allowing for textual input and output. Eventually getting more complex and allowing for images and graphics to be displayed and input.

This was the process that created the modern computer.

Now these modern computers could store near infinite data in efficient storage. Could calulate near infinite amounts of formulas. And could provide outputs that humans could ineract with in the modern world.

So we can think of a modern computer as a device which can store data, can store massive amounts of data, can process massivley complex algorithms with this data, and can output it in formats that humans can consume.