Adding numbers with gears

Now-a-days we use digital calculators to add big numbers. But in earlier days, mechanical calculators did this job instead. Blaise Pascal built the first mechanical calculator in 1642. Mechanism he built can be understood with the help of this simplified prototype. We will use 3D printed gear with 10 teeth, marked with numbers 0 to 9. With the help of a pin, we can place it on the cardsheet. Let us place another gear on the left side of the yellow one. Push pins are placed next to zero digit and mark the starting or reset position. As you know, place value is the value of each digit in a number, based on its position. Yellow gear represents Unit Place, Green one tens place. For number 13, digit 1 has a place value of 10 while digit 3 has a place value. Number 13 can be represented like this while 54 like this. But how can these gears help us in addition ? Let us connect these two gears with another one. This gear is also known as an accumulator. It rotates with the yellow gear all the time but moves the green gear only one digit forward for each complete rotation of the green gear. This is also known as carry operation. How about adding two numbers : 13 and 15 ? Let us rotate yellow gear till digit 3 is placed next to the pointer. Let us rotate the green one till digit 1 is placed next to the pointer. In order to add 15, we have to move the yellow gear 5 times in a clockwise direction and the green gear once. Now you get the result of addition 28. Let us add another number 45 to this . I will rotate the yellow gear 5 times clockwise as 5 is in the Units place . Green gear needs to be rotated 4 times as 4 is in the tens place. You guessed it right. 73 is the result of adding 28 and 45. Let us reset the gears to the initial position. How about adding 67 and 55 ? Result will be greater than 100. That means we need three gears to represent the same. Let us add white gear. It represents hundreds place. We will also place this gear to connect tens place and hundreds place. This gear will move white gear one digit forward for one complete rotation of green one. Let us rotate yellow gear clockwise till digit 7 is at the pointer and green one till digit 6 reaches the pointer. Now to add 55 to this number , we will move yellow gear 5 teeth and green one five teeth. Our result is 122. Accumulator moved green gear one teeth forward when yellow gear rotated ones. Same was true with white gear as well. How about adding 596 to this number. Let us move yellow gear 6 teeth, green one 9 teeth and white one 5 teeth. And the result is 718. For larger numbers, we can add more accumulators and gears on the left side. How about subtraction ? Let us reset all the three gears. How about subtracting 23 from 37 ? Let us represent 37 first. To subtract 23, we will move the gears anti-clockwise. Yellow gear three teeth anticlockwise and green gear 2 times. Result is 14. Let us try bigger number 786 - 139 647. With this arrangement you can do additions as well as subtractions. These gears were designed in freecad and printed on Ender 3. I will create another video for the same. Can you think of alternatives if you don’t have a 3 d printer ? Give it a try . Thank You. Link to Freecad and PDF file is here. #mathematics #gears #addition #3dprinting