so today for my project I'm going to be talking about the Trachtenberg system and this is done by Bojana vazhi as my extra credit so first what you need to know about this system is it involves direct multiplication and it's about two-digit multipliers we'll get into that in just a second first thing we need to talk about our leading zeros we need to put as many leading zeros in front of the first multiplier as there are in the second digit so to give you an example of what a leading zero looks like what we're gonna do is use 330 times 293 you can see in 293 there are three digits those three digits will be resembled as zeros and it's going to go in front of 330 so what we'll do is we'll put 0 0 from 0 and that's the 3 digits and then we'll put 330 we'll all multiply this by 293 you can see that the second digit does not change at all for the second example we'll do just a smaller number that are two digits so we'll do 25 multiplied by 17 we'll take the two digits of 17 and we'll put them in front of the 25 so it'll be 0 0 25 times 17 from this point you can see it's actually pretty basic so before we get into it there's ways that you need to think we should be thinking on paper and we should also be thinking in our mind so we're gonna write down all the numbers on paper but we also need to do the mental math in our head so here we are at our first example you can see it's 23 times 10 we have already taken the zeros that are taken by the two digits in the tenth and we place them in front of the 23 our red marker right here is going to resemble the one multiplying nothing and the first step the first digit will always be in between this digit the multiplication symbol will never be multiplying anything in the first step our green marker which is going to resemble the zero is going to be multiplying 3 so what we're gonna have is zero times 3 and the next step you'll see how our red marker our one has moved to multiply 3 everything moves down one digit at a time our zero which is our green marker has moved to the 2 this is going to carry on as well for the next step the 0 is going to multiply the 0 for the green marker and the red marker is going to move down from the 3 previously to now the 2 and in our last step the red marker is going to go from the 1 and it's going to multiply the 0 and our green marker is going to go from the 0 and multiply all the way to the next 0 so moving on from our first example which just so shows the basic lines that we need to do for the multiplication we're gonna go and check out a real example so this is going to be 12 times 23 so the first step what we're gonna do is we're gonna take the two digits of zeros and put them in front of the 12 so we'll have 0 0 1 2 times 23 our first step is going to be taking the 2 and we'll take the red marker and draw it to nothing and then we'll have a 3 and we'll take our marker and draw it to the 2 so on the side we can do 3 times 2 2 and that's going to equal 6 and then we're going to have 2 times 0 which is going to equal 0 this when you add it is going to end up to 6 we'll put a line underneath this and we'll put the 6 underneath the 2 so then we can move on to our next step we'll have 0 0 1 2 times 23 and we'll redraw our lines now the two is going to connect to the 2 and R 3 will connect to the one we can go ahead and draw our line and we'll do two times two that's going to be equal to four and then we'll have 3 times 1 that's going to equal to 3 when we add it together it's going to equal 7 so now we can rewrite the 6 that we already found previously and then we can bring in the 7 which we just found now we're going to rewrite this one more time 0 0 1 2 times 23 we'll grab our red marker and instead of going to 2 this time we'll move down a digit to the 1 and the green is going to go all the way to the 0 so at this point we can draw the line we can come back and do our math on the side if we do 3 times 0 that's going to equal 0 and if we do 2 times 1 that's going to equal 2 we add that together you'll see that we get 2 again or just 2 in general so then we'll do 6 7 which we're just copying down from here and our 2 is gonna go underneath the 0 for our next step we're gonna just finish it out we're gonna write 0 0 1 2 times 23 again we'll draw our line under 0 0 1 2 so now we're going to go ahead and draw our lines again instead of our two going to the 1 the 2 is gonna go to this first 0 and the 3 is gonna go to the second 0 we can come to the side and do the math again 2 times 0 is equal to 0 3 times 0 is equal to 0 if you add those together it's all 0 so we'll copy down the 2 7 6 and we can add the new 0 in front so by taking this 12 times 23 is going to be equal to 276 so let's make the next example a little bit harder we're going to introduce a three-digit first number but we still have a two-digit second number we'll set it up the same two digits and 41 so we'll do zero zero for the two digits two hundred eleven times forty one we can go ahead and draw our our first line our forty one is going to go to oblivion and our one is gonna go to this one right here so we can do four times nothing so 4 times 0 which is equal to zero second one is 1 times 1 1 times 1 is equal to 1 draw a line under everything and our first answer that we found is going to be 1 and we'll put that right here ii will rewrite it again 0 0 to 11 times 41 put the line under we'll take our red and now our Reds gonna go to the 1 and our green is gonna go to the next one we'll do 1 times 1 for the green and equals 1/4 times 1/4 the red and that equals 4 add them together we'll get 5 so we'll bring one back and bring the 5 over we'll do this again 0 0 to 11 times 41 draw the line under we'll take the red bring the 4 to the next one we'll take the green and bring it to the 2 if we do the math 1 times 2 is equal to 2 and 4 times 1 is equal to 4 add it together and you'll get 6 so we can bring the 1 down the 5 down all from up here and then we'll bring the 6 underneath the two well write it again 0 0 to 11 times 41 put a line under take the red from the 4 and now it's going to go to the 2 and the green is go all the way to the first zero do the math one times zero equals zero four times two equals eight add that together and I will get eight so we'll bring down the 1 we'll bring down the five the six and we'll bring the eight over underneath the first zero in the last time we'll write it down times 41 draw the line grab the red marker from the four we're gonna go all the way to the first zero and the green from the one is gonna go all the way to the second zero I found the step not to be necessary but it's worthwhile showing so four times zero for our red is going to equal zero one times zero from the green it's going to equal zero total zero and we'll copy down this number eight six five one 8650 one we can add the zero in front if you want just to show that it does equal this so 211 times 41 is equal to eight thousand six hundred fifty-one and to double-check our math just like we do on test what we're gonna do is do 211 times 41 and bam there's our answer so thank you for watching and I hope you enjoyed learning how to use this system