Yeah ikr, I love how people say "they dojt teach this in the schools" where they have been taught about this since 6th grade itself. Took me only a few seconds, when he said "twice the number"
Textbooks and homework is what students had prior or before podcasting was available. The results reflect the difference in performance by the students!.
Not at all. School system has a different purpose which is to teach you the concepts. Whereas, these shortcuts which are a distilled form of those concepts serve a different purpose, which is to make your calculations faster. Both have their place.
I am 70 years old and grew up loving math and doing math games and quizzes with my father. I miss math class. These problems had me smiling and getting them right...”Come on! Give us another one!” 😊 Thanks! BTW, I’m in South Carolina, USA
Very useful video, This method would have never crossed my mind in a million years and I would have been stuck doing it the hard way in my finals, But now I'm armed with an easy and surprisingly fun method. Thank you a lot
Those who say "I can't do math" has never seen these alternate techniques. Not only are they efficient but a lot of fun to play around with. Wish schools taught this. Even if it's after learning the basic fundamentals first.
Those who say "I can't do math" have never seen these techniques because they don't enjoy doing maths. They also wouldn't think this is fun to play around with.
I'm rubbish at maths but find this easy to follow, unlike when I was in school and regularly had the board duster thrown at me by the maths teacher for getting it wrong!! Thank you
Not only can you extend this to any number of digits, but the 3rd power stuff, while not likely something you can work out manually, is very interesting. It basically adds another thing to multiply in each of the steps: the number you are trying to square with various digits 0'd out. Unique to 3rd powers. For example, say you wanted to calculate 84345458^3 using this method. The second step in the other method has you multiply digits 1 and 3, 8*3 here. Then you need to multiply that by 80345458 and then finally multiply all that by 3, instead of 2. The 0's in the number are always the digit/digits you skip over in each step. This 80345458 number is completely absent in 2nd powers.
I tried it on numbers with 4 or more digits and this technic works there as well. As explained in another comment, this is just another way to visualize (a+b)^2. ex: 4567^2 16253649 : a^2, b^2, c^2, d^2 406084 : 2ab, 2bc, 2cd 4870 : 2ac, 2bd 56 : 2ad = 20,857,489
I love your videos, now I see why I had a rough time with math all these years. Even though I scored A's in Science, Health, History, English, Social Studies et. al.
You are the Mister Rogers of math, Josh! Thank you for your gracious style and clear instruction. It is a genuine pleasure to listen to your lessons. I have learned so much from you. I have suffered for decades with math anxiety to the point of mental paralysis. Thanks to your teaching vids, I'm doing math in my head quickly and accurately. I appreciate the many new ways of thinking about numbers and mathematical operations that you bring to the table. Keep up the great work!
It's actually possible to extend this to numbers with 4 digits as well. If we take number like (512)² and write it like abc² your pyramid looks like a² b² c² 2ab 2bc 2ac like he showed us in the video. But if you have a number like 1234² you can do this as well. Your pyramid should then look something like this: a² b² c² d² 2ab 2bc 2cd 2ac 2bd 2ad. Really cool!
Yep, this is because the square of a sum is the sum of: • the squares of each individual component, and • twice the pairwise products of non-matching components Eg (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc. (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd.
This channel makes me happy because now, if I learn these methods, I can be successful at everyday maths, instead of being anxious and very slow. Thank you!
Great. Will keep practicing mental calculations. This greatly simplifies squares, yet still aiming at 1 min for 3-digit ones. Awesome practice. Thank you.
3:38, analytical proof: Let's "x" be the two digit num which is = (10a + b) where a and b are the corresponding digits of that number. Thus, x = (10a + b) Now let's square it, => x² = (10a + b)² Therefore, x² = (100a² + b² + 20ab) ---(i) Clearly, from (i) you can already see that, it is nothing but the sum of the squares of '10a', 'b' and, the twice the product of '10a' and 'b' i.e. x² = (10a + b)² = [ *(10a)²* + *(b)²* + *2(10a)(b)* ] Now you'll notice that you had to take 10 times of 'a', that's because a is in the Ten's place, for example 12 = 1×10 + 2, just like that. so upon squaring you'll obviously get a 4 digit number, let's see how! w.k.t, if x be a two digit number, then: x² = *(100a² + b²) + 2(10a × b)* a = Ten's digit, b = Unit's digit. Let's take x = 69 for fun. Now let's find out it's square. 69² = 100×36 + 81 + 2(10×6 × 9) = 3600 + 81 + 2(10×6 × 9) = *3681 + 2(6×9)×10* = *3681 + 56×10* Or, *3 6 8 1* +0 *5 6* 0 ------------------ 4 2 4 1 ← yes that's the answer! (10×[{(420)}] + 41 :P jk) Now ↑ above you'll notice that 3681 is nothing but 36 or 6² and 81 or 9² !!! And also, we get "0560" below 3681 because 560 is in hundred's place so it shits to the right side, but 560 also have a "0" in the last, so we see that the number "56" stays right below 68 in 3681 and that's what happens everytime with all two digit number squared!!! And now you know exactly why it happens :)
You sir are an excellent teacher. I dropped out of high school before finishing and later in my late 30's I was disabled and attended university on a rehab scholarship. Having not been exposed to anything beyond simple addition, subtraction, multiplication and division, I struggled with algebra, trig, and had to start out with remedial or high school math. If I had had a teacher as yourself I would have finished with a perfect 4.0 GPA as math is the only subject I didn't get straight A's in. Because of my lack of math background I had to settle for a 3.85 GPA.
(a + b) ^2 == a^2 + 2ab + b^2 (with 2 figures). (a + b + c)^2 = a^2 + 2ab +2bc + 2ac + b^2 + c^2 (with 3 figures). Although I knew these formulas, I had not figured out how to do this calculation in practice: thank you! I used to have another method that you can use for numbers as long as you know some of the lower squares by heart. If x^2 is the square you need, you use (x - a)(x + a) + a^2. I call it the harmonica because it works like a harmonica. It works like this. You have f.e. 17 = 20 x 14 + 3^2 = 289. Btw, this trick I got from my granddad (an illiterate fisherman, but he knew squares). I will admit, not as easy to use as your trick for larger number, because you end up with a smaller square. So doing 407 would be a little harder: 407^2 = 400 x 414 + 7^2 = 4 x 41400 + 49 = 165600 + 49 = 165649. Not as easy, but still, I used it for a long time. thanks for the tips and the video!
Good one mate. I was helping my grade 6 daughter with 2 digit squares and said look, this is (x+y) squared, so x squared + y squared + 2xy. This makes it so much more straight forward. I must check out more of your work. Being 77, I hate these modern maths, simple maths made complex.
Hello..new subscriber here...just stopped to thank you..I have an important government exam coming up and was very very nervous about the calculation part and I can't really get the correct answer quickly..these tricks will help a lot.. thank you so much sir ❤️
Thanks for the very clear instructions and showing how it’s done simply. I’ve now picked 50 random examples I intend to use these as a tool to learn better. I’m hoping this educates and helps me better my knowledge and grasp it. Thank you.
632²=360000+900+4+36000+2400+120=399424 there ya go. It's an old mathematical trick. Not hard at all. It's so old that I bet all of you actually knew it before the video, without realizing it. (a+b) × (a+b) = a²+2ab+b² It holds true no matter which base you use. And let's show an example of 23² = 20² + 2×20×3 + 3² or 400+120+9 aka 529. It's how everyone turns big multiplications into smaller ones. 322*345 is difficult but 300*345 + 20*345 + 2*345 looks a lot easier and 300*300 + 300*40 + 300*5 + 20*300 + 20*40 + 20*5 + 2*300 + 2*40 + 2*5 looks even simpler to do. And for the sake of answering that it's 90000+12000+1500+6000+800+100+600+80+10 or 111090. It's exactly the same principle.
@@livedandletdie Feeling a dimmy wit at the 2ab stage as to me that is doubling and not squaring. Maths always was my weakest subject and still is along with needlework!!! xxxx
I'm 73 and left school in year 9. I failed math in the three years of high school that I did. I wish I'd known all these ways of doing math easily way back then, I think I would have blitzed it. Thanks for making it so much fun. LOL now I can't get enough of them. 😁
Im 32 and I was almost convinced that ill never ever pass in any math quiz until i found this channel.. Thank you so much..❤️❤️❤️ Hoping to see more and more of your math tricks..💕💕💕
The order of the calculations is important as it relates to the tens column that the answer should be in. For example 60*60=3600 2*(60*7)=840 7*7=49. However by using commas to denote the separation between the powers of 10 (lets call them open numbers) it enables one to handle the multiplication of any number of digits provided the correct sequence of numbers that are multiplied together are added together. In this case it would be 6*6=36, 2*(6*7)=84, 7*7=49 giving the open number result 36,84,49. From this it can be seen that the compressed number answer in Hindu-Arabic notation is 4489. The problem comes from teaching long multiplication in columnar form and only allowing each column to contain one digit. This requires one to handle the carry (or compression of the open number result) in the middle of all the multiplication operations. If one turns the multiplication sum on its side and uses open numbers the process suddenly becomes a lot easier and quicker to perform as one can extend the power of 10 number calculations to handle results with more than one digit. This has the benefit of leaving the compression of the final open number result to the end of the process. Thus the order of operations can be done either right to left (traditional) or left to right (for mental speed maths) provided one is clear about which digit multiplications for each power of ten computation should be added together. For example Units: 7x7 = 49 Tens: 6x7+7x6 = 84 100's: 6x6 = 36 result: [36,84,49] = 4489
Good trick, but you don't have to stop at 1000. To square a four-digit number: The first row has the squares of each of the four digits, as with the three-digit numbers. For the second row, multiply the adjacent digits and double the result, again as with the three-digit numbers. For example, with 2357, the second row becomes 061535. The only minor complication comes when any result is more than 100, in which case remember to carry the 1. For the third row, multiply the first and third digits and double the result, then multiply the second and fourth digits and double the result. Again, remember to carry the 1 whenever a result exceeds 100. For the (new) fourth row, multiply the first and last digits and double the result. This technique can generalized for squaring any number with n digits. In the spirit of all math textbooks, this is left as an exercise for the student. 😀
@@arpanmascarenhas1048 Just remember that each new number goes into the next two positions. With a three-digit number, think of the third digit as overflow. For example, to square 6789: The first row is 36496481, the square of each digit. For the second row, start with 6×7×2=84. Add to that 7×8×2=112, when, with two spaces reserved for this new number, you now have 8512 in this row. Then add 8×9×2=144 the same way, making this row 851344. I'll add a zero to the end to make the numbers align more clearly, making the second row 8513440. The third row starts with 6×8×2=96. Then add 7×9×2=126, giving 9726. Add two zeroes for alignment, making this row 972600. For the fourth row, you have 6×9×2=108 plus three zeroes for alignment, giving 108000. Adding them up, you get 36496481 + 8513440 + 972600 + 108000 = 46090521, and you're done.
Hi Josh, thanks for an easy explanation on squaring numbers up to 1000. I don't recall this type of explanation in any of my early schooling or tuition, which I shall now say it was knowledge denied. For this expanation, you deserve an A+🌟🌟🌟🌟🌟👏👍🥇.
@@Itsme7384 It actually does work for 999, you just have to contract the number. If you did this correctly, you get 162162 in the middle. What you have to do is add the two numbers in the middle to get 16362. The catch is that you have to line it up with the 818181. At the end you get 818181+163620+162 = 998001 which is 999^2. The same applies to 696.
@@Itsme7384 there's an easier method there. The concept being used in the video is to expand it as a binomial or a trinomial and just using the appropriate number of zeros. There's another trick: use a-b for numbers close but lesser than multiple of 10, 100. Like 696^2 = (700 - 4)^2... So 490000 + 16 - 5600 = 484416 999^2 = (1000 - 1)^2... This becomes 1000000 + 1 - 2000 = 998001
The last digit of the 3 digit number would go in the same position as the last digit when it is 2 digits. So the first 108 would go directly under 368 and the second 108 would go below the 813. That way the last digits are still under the 8 and the 3.
Yeah exactly! (a+b)² = a² + 2ab + b². I'd rather do it like this though 512² -> (500+12)² = (500)² + 2(500)(12) + (12)² = 250,000 + 12,000 + 144 = 262,144 It's kinda easier to digest. But either way works.
Had to look at this and figure it out...it gets sloppy. You put the two numbers together but add the one's digit from 162 to the hundred's digit of 144 together getting 16344 and like standard math this goes below the first number shifting one to the left (leaving a space under the 1's digit) the other 144 in the third row is once again shifted another row leaving an empty space under the 10's digit. It might be easier to figure out than the standard squaring method, but I don't think I could do this in 5 seconds in my head.
I decided to learn math as an adult because i have nothing better to do with my free time.... plus I regret not taking math seriously as a kid. These methods are helpful.
Hey... this scales up infinitely. Let’s test it on 62,347 Finding the squared numbers is simple... You get: 3,604,091,649 Then you just add the numbers multiplied together: 241,224,560 36,164,200 4,828,000 840,000 To get the answer: 3,887,148,409 But at this point, you’re better off using the four square method instead. The four square method is basically the same thing but in a chart.
This is really helpful. I tried doing another one on my own. i was going for 596 squared. but i found it was tricky to do 9x6x2 because you end up with 108 which makes that line 5 digits. and im not sure how to put it in the "middle" the 60 since theres 5 digits. Thanks for your help
I saw it question and tried it and it has me puzzled as well... I did it the old way and got the answer though and I've been looking at it since an can't come up with a solution. Lets see if he replies
@@didididtuiu4494 um this doesn’t make any sense? Where did the 5 come from. 816449 114112 seems like it would be correct, so please actually explain as we don’t know how you got the 5 and where the first 1 went in 114.
hey mate, love it learnt it pretty easy but i just did good old 999^2 but you see when we start to multiply 9x9 then double it, what do we do with the 3 digits. (the ones you showed equal 2 digits which is easy)
discovered this as a kid. and once you understand it you can do heavy multiplication using it as well. we had a vedic math demonstration when I was in 6th class and the guy showed as the same trick however demonstration was short and he did't give much explanation. the trick is to understand position of each multiplication by position of digit so AB X CD : (10A+B) X (10C + D) = 100AC + 10(AD + BC) + BD i.e. (reading right to left) BD gives 1st digit with a carryover, AD + BC + carryover gives 2nd digit + new carryover , AC + the new carryover gives 3rd and 4th digit now squaring changes simplifies the formula also when you start practising this you realise how to use 0 to ignore some calculation and use 1 to simply write in some products another trick in squaring ( in multiplication in general) is if squares of a 2 digit subpart ( in extension multiplication of 2 digit numbers) are known to you , you can use it to simplify it . e.g. 512^2 take this as (500+12)^2 going right to left 12^2 = 144 i.e. 1st & 2nd digit = 44 and 1 carryover , 2 * 5 * 12 = 120 120 + carryover 1 gives 3rd and 4th digit 21 and 1 as carryover , 5^2 = 25 is 5th and 6th digite + carryover 1 = 26 so , 26 21 44
I am a math teacher from Egypt, and I and my pupils benefited from your methods of explaining mathematics and I would like to thank you a lot
You should trade your job with him. 😬
اجدع ناس
I already did that.
@@mytravls :|
تحيا مصر و المدرسين المصريين. 🇸🇦❤️🇪🇬
It took me a solid three minutes to realise that this is basically (a+b)^2 = a^2 + 2ab + b^2. A very creative way (and yet simple) of applying this.
wow wouldn't of picked this up :)
:OOO didnt even realise
Yeah ikr, I love how people say "they dojt teach this in the schools" where they have been taught about this since 6th grade itself. Took me only a few seconds, when he said "twice the number"
no it isnt
@@scottcaine4514 yes it is
I love how excited and proud he sounds.
Why does the comment says '10 years ago' '3 days ago' wait wha-
Wait what
@@lfn4p829 It's... his name
@@lfn4p829 I wrote the comment 10 years ago, and edited 3 days ago.
@@BenDreemurr44 lol , I love your sarcasm
Didn't realize the thing i needed in life was an Aussie man showing satisfying math hacks
dat choice of words.
He sounds like he's always on the verge of laughing
🤣
😭😭😭
no cap lol
💀💀💀
Wow this fact made me laugh
Imagine this guy is your math teacher how much better would this be .
True
Textbooks and homework is what students had prior or before podcasting was available. The results reflect the difference in performance by the students!.
😂🔥
Not at all. School system has a different purpose which is to teach you the concepts. Whereas, these shortcuts which are a distilled form of those concepts serve a different purpose, which is to make your calculations faster. Both have their place.
No.
I am 70 years old and grew up loving math and doing math games and quizzes with my father. I miss math class. These problems had me smiling and getting them right...”Come on! Give us another one!” 😊 Thanks! BTW, I’m in South Carolina, USA
got me by 11 years LOL but I was doing the same sitting here in Tianjin in China at 4 am LOL
@@RBLXGiveaway How is that an unreasonable age?
@@RBLXGiveaway why?
Lack of reply from Amy might imply she indeed Was 70
@Ed EdwardsThat’s apart of the American dialect. It is not a typo
It’s ridiculous how much this channel improved my mental math in so little time
Pls solve 798² bro
I didn't get it.
@@tudorftbl but how dude
@@unpluggedalphaa
792^2
498164
12744
112
636804
quite easy but i know what you might have done wrong, you wrote 126144 instead of 12744 right?
@@KGR_ 😀 thank you brotha
Im 10½ and i know every square up to 1000, it just took me time to tell someone wuick when they asked. This helped improve my speed!
You truly have a gift for teaching and numbers... A very rare combo mate!! 👍
Thanks mate
@@tecmath 👍
That's not a gift,that's simply (a+b)^2
@@arwen3478 you use (a+b+c)^2 lmao whats wrong
@@karenamma7716 you wont be able to tell that if this video havent told you about this in the first place lul
Very useful video, This method would have never crossed my mind in a million years and I would have been stuck doing it the hard way in my finals, But now I'm armed with an easy and surprisingly fun method. Thank you a lot
Those who say "I can't do math" has never seen these alternate techniques. Not only are they efficient but a lot of fun to play around with. Wish schools taught this. Even if it's after learning the basic fundamentals first.
Those who say "I can't do math" have never seen these techniques because they don't enjoy doing maths. They also wouldn't think this is fun to play around with.
They do teach this people are just too dumb to apply it to real life this is (a+b)^2 it’s taught in 6th grade in most places
I arrived at this trick but you picturised it first ,BRAVO CONGRATULATIONS
I love your positivity, it makes it so much more fun to learn math concepts!
You consistently blow my mind mate. Mathematics is such a beautiful subject and I find this so appealing. Thank you for all you do
I'm rubbish at maths but find this easy to follow, unlike when I was in school and regularly had the board duster thrown at me by the maths teacher for getting it wrong!! Thank you
Hi Josh. Just wanted to say a big thanks for your vids. My son and I love the clear, simple explanations that make maths so much easier. 👍. Matt
Not only can you extend this to any number of digits, but the 3rd power stuff, while not likely something you can work out manually, is very interesting. It basically adds another thing to multiply in each of the steps: the number you are trying to square with various digits 0'd out. Unique to 3rd powers. For example, say you wanted to calculate 84345458^3 using this method. The second step in the other method has you multiply digits 1 and 3, 8*3 here. Then you need to multiply that by 80345458 and then finally multiply all that by 3, instead of 2. The 0's in the number are always the digit/digits you skip over in each step. This 80345458 number is completely absent in 2nd powers.
Thanks , it helped me a lot for finding the square of a number
I tried it on numbers with 4 or more digits and this technic works there as well. As explained in another comment, this is just another way to visualize (a+b)^2.
ex: 4567^2
16253649 : a^2, b^2, c^2, d^2
406084 : 2ab, 2bc, 2cd
4870 : 2ac, 2bd
56 : 2ad
= 20,857,489
now you can square numbers up to 10000
also, ive found a pattern here:
aa, bb, cc, dd (0 offset)
ab, bc, cd (1 offset)
ac, bd (2 offset)
ad (3 offset)
so, 5 digits would be
aa, bb, cc, dd, ee (0 offset)
2ab, 2bc, 2cd, 2de (1 offset)
2ac, 2bd, 2ce (2 offset)
2ad, 2be (3 offset)
2ae (4 offset)
12345^2 = 152399025 (from calculator)
0104091625
04122440
061630
0820
10
0152399025 (from method)
I love your videos, now I see why I had a rough time with math all these years. Even though I scored A's in Science, Health, History, English, Social Studies et. al.
You are the Mister Rogers of math, Josh! Thank you for your gracious style and clear instruction. It is a genuine pleasure to listen to your lessons. I have learned so much from you. I have suffered for decades with math anxiety to the point of mental paralysis. Thanks to your teaching vids, I'm doing math in my head quickly and accurately. I appreciate the many new ways of thinking about numbers and mathematical operations that you bring to the table. Keep up the great work!
It's actually possible to extend this to numbers with 4 digits as well. If we take number like (512)² and write it like abc² your pyramid looks like a² b² c² 2ab 2bc 2ac like he showed us in the video. But if you have a number like 1234² you can do this as well. Your pyramid should then look something like this: a² b² c² d² 2ab 2bc 2cd 2ac 2bd 2ad. Really cool!
Can you explain the last part in pyramid form?
works with 5 digits too
Yep, this is because the square of a sum is the sum of:
• the squares of each individual component, and
• twice the pairwise products of non-matching components
Eg (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc.
(a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd.
He used (a+b)² it is simple ..but the way he makes it easy to understand that is soo great..❤️❤️❤️❤️
Thank you so much for breaking this down gently and positively. I love mental math and your channel is really fun.
This man sounds like he’s on the verge of yawning when speaking
for me it feels likes he's on the verge of laughing out loud
Most accurate observation ever...
Now I'm yawning when I hear him speak. 🤭
For me it sounds like he´s about to giggle all of the time
I always enjoy sharing these with my class. Big thumbs up!
yea its fun when they look at u how fast u have done this but actually we r using these tricks to do faster .... :)
This channel makes me happy because now, if I learn these methods, I can be successful at everyday maths, instead of being anxious and very slow. Thank you!
Sir I really can't express in words how helpful your videos are . Respect
Great. Will keep practicing mental calculations. This greatly simplifies squares, yet still aiming at 1 min for 3-digit ones. Awesome practice. Thank you.
3:38, analytical proof:
Let's "x" be the two digit num which is = (10a + b) where a and b are the corresponding digits of that number.
Thus, x = (10a + b)
Now let's square it,
=> x² = (10a + b)²
Therefore, x² = (100a² + b² + 20ab) ---(i)
Clearly, from (i) you can already see that, it is nothing but the sum of the squares of '10a', 'b' and, the twice the product of '10a' and 'b'
i.e. x² = (10a + b)² = [ *(10a)²* + *(b)²* + *2(10a)(b)* ]
Now you'll notice that you had to take 10 times of 'a', that's because a is in the Ten's place, for example 12 = 1×10 + 2, just like that.
so upon squaring you'll obviously get a 4 digit number, let's see how!
w.k.t, if x be a two digit number, then:
x² = *(100a² + b²) + 2(10a × b)*
a = Ten's digit, b = Unit's digit.
Let's take x = 69 for fun.
Now let's find out it's square.
69² = 100×36 + 81 + 2(10×6 × 9)
= 3600 + 81 + 2(10×6 × 9)
= *3681 + 2(6×9)×10*
= *3681 + 56×10*
Or,
*3 6 8 1*
+0 *5 6* 0
------------------
4 2 4 1 ← yes that's the answer! (10×[{(420)}] + 41 :P jk)
Now ↑ above you'll notice that 3681 is nothing but 36 or 6² and 81 or 9² !!!
And also, we get "0560" below 3681 because 560 is in hundred's place so it shits to the right side, but 560 also have a "0" in the last, so we see that the number "56" stays right below 68 in 3681 and that's what happens everytime with all two digit number squared!!!
And now you know exactly why it happens :)
You sir are an excellent teacher. I dropped out of high school before finishing and later in my late 30's I was disabled and attended university on a rehab scholarship. Having not been exposed to anything beyond simple addition, subtraction, multiplication and division, I struggled with algebra, trig, and had to start out with remedial or high school math. If I had had a teacher as yourself I would have finished with a perfect 4.0 GPA as math is the only subject I didn't get straight A's in. Because of my lack of math background I had to settle for a 3.85 GPA.
Thanks for the mini-biography
(a + b) ^2 == a^2 + 2ab + b^2 (with 2 figures). (a + b + c)^2 = a^2 + 2ab +2bc + 2ac + b^2 + c^2 (with 3 figures). Although I knew these formulas, I had not figured out how to do this calculation in practice: thank you! I used to have another method that you can use for numbers as long as you know some of the lower squares by heart. If x^2 is the square you need, you use (x - a)(x + a) + a^2. I call it the harmonica because it works like a harmonica. It works like this. You have f.e. 17 = 20 x 14 + 3^2 = 289. Btw, this trick I got from my granddad (an illiterate fisherman, but he knew squares). I will admit, not as easy to use as your trick for larger number, because you end up with a smaller square. So doing 407 would be a little harder: 407^2 = 400 x 414 + 7^2 = 4 x 41400 + 49 = 165600 + 49 = 165649. Not as easy, but still, I used it for a long time. thanks for the tips and the video!
That’s nice. Been using it for a long time myself. You can do larger numbers if you adapt it slightly - see my videos on my channel if you would like!
Nice method
Good one mate. I was helping my grade 6 daughter with 2 digit squares and said look, this is (x+y) squared, so x squared + y squared + 2xy. This makes it so much more straight forward. I must check out more of your work. Being 77, I hate these modern maths, simple maths made complex.
Hello..new subscriber here...just stopped to thank you..I have an important government exam coming up and was very very nervous about the calculation part and I can't really get the correct answer quickly..these tricks will help a lot.. thank you so much sir ❤️
you are like my elementary school math teacher long long time ago...make the students enjoy math...thanks again 👏👌
This is essentially how you multiply numbers on paper normally, with a couple shortcuts to make it easier. Nice job i like this :)
this guy taught me more mental math in 15 minutes of his content than my school did in 13 years of maths class
I love how hes so happy about telling us about this, makes learning it a whole lot easyer
I like the way you work out these equations. I work the numbers on the side and add them. I have become hooked on working these math problems.
Loved it. I really enjoyed the video. I hope you stay safe.
Very helpful mathematics teacher. Keep going up👍
That. Is. Great.
I haven't enjoyed math since the 60's! If I'd had that, I might have made it to algebra...
Honestly helping me so much as I am practicing for my ASVAB for the marines
Thanks for the very clear instructions and showing how it’s done simply. I’ve now picked 50 random examples I intend to use these as a tool to learn better. I’m hoping this educates and helps me better my knowledge and grasp it. Thank you.
omg I was practicing squaring 3 digit numbers and as soon as I wrote down 632 i heard him say 362, it was so close!
My call was 367
632²=360000+900+4+36000+2400+120=399424 there ya go. It's an old mathematical trick. Not hard at all. It's so old that I bet all of you actually knew it before the video, without realizing it. (a+b) × (a+b) = a²+2ab+b² It holds true no matter which base you use. And let's show an example of 23² = 20² + 2×20×3 + 3² or 400+120+9 aka 529.
It's how everyone turns big multiplications into smaller ones.
322*345 is difficult but 300*345 + 20*345 + 2*345 looks a lot easier and 300*300 + 300*40 + 300*5 + 20*300 + 20*40 + 20*5 + 2*300 + 2*40 + 2*5 looks even simpler to do.
And for the sake of answering that it's 90000+12000+1500+6000+800+100+600+80+10 or 111090. It's exactly the same principle.
I was stunned I got 362 as my guess!!!! xxxx
@@livedandletdie Feeling a dimmy wit at the 2ab stage as to me that is doubling and not squaring. Maths always was my weakest subject and still is along with needlework!!! xxxx
@@janemccourt5022 My needlework needs improvement, thankfully my maths is very good.
Thanks, man! You've made my math practice easier!😀
I'm 73 and left school in year 9. I failed math in the three years of high school that I did. I wish I'd known all these ways of doing math easily way back then, I think I would have blitzed it. Thanks for making it so much fun. LOL now I can't get enough of them. 😁
Im 32 and I was almost convinced that ill never ever pass in any math quiz until i found this channel.. Thank you so much..❤️❤️❤️ Hoping to see more and more of your math tricks..💕💕💕
Watching these videos is far more convenient than reading books about Vedic arithmetic. Cheers!
Had to make sure this wasn’t an April’s fools video lol thank you so much!
Lol
Sorry to reply lateeeeee
Love the ones with the "0" in the middle.
We need a techmath mental math app
Yes!
Just started my pre electrical apprenticeship and your videos are very helpful!
I hate math and your making me Love it. Thank you ❤
(a+b)^2=a^2+b^2+2ab. SIMILARILY,
67^2 can be written as (60+7)^2= 60*60+7*7+2*60*7= 4489
The order of the calculations is important as it relates to the tens column that the answer should be in. For example 60*60=3600 2*(60*7)=840 7*7=49.
However by using commas to denote the separation between the powers of 10 (lets call them open numbers) it enables one to handle the multiplication of any number of digits provided the correct sequence of numbers that are multiplied together are added together. In this case it would be 6*6=36, 2*(6*7)=84, 7*7=49 giving the open number result 36,84,49. From this it can be seen that the compressed number answer in Hindu-Arabic notation is 4489.
The problem comes from teaching long multiplication in columnar form and only allowing each column to contain one digit. This requires one to handle the carry (or compression of the open number result) in the middle of all the multiplication operations. If one turns the multiplication sum on its side and uses open numbers the process suddenly becomes a lot easier and quicker to perform as one can extend the power of 10 number calculations to handle results with more than one digit. This has the benefit of leaving the compression of the final open number result to the end of the process. Thus the order of operations can be done either right to left (traditional) or left to right (for mental speed maths) provided one is clear about which digit multiplications for each power of ten computation should be added together. For example
Units: 7x7 = 49
Tens: 6x7+7x6 = 84
100's: 6x6 = 36
result: [36,84,49] = 4489
@@sharonjuniorchess Thankyou for this effort !
Appreciated!
Good trick, but you don't have to stop at 1000. To square a four-digit number:
The first row has the squares of each of the four digits, as with the three-digit numbers.
For the second row, multiply the adjacent digits and double the result, again as with the three-digit numbers. For example, with 2357, the second row becomes 061535. The only minor complication comes when any result is more than 100, in which case remember to carry the 1.
For the third row, multiply the first and third digits and double the result, then multiply the second and fourth digits and double the result. Again, remember to carry the 1 whenever a result exceeds 100.
For the (new) fourth row, multiply the first and last digits and double the result.
This technique can generalized for squaring any number with n digits. In the spirit of all math textbooks, this is left as an exercise for the student. 😀
carry the 1 where though? To the next row?
@@arpanmascarenhas1048 Just remember that each new number goes into the next two positions. With a three-digit number, think of the third digit as overflow. For example, to square 6789:
The first row is 36496481, the square of each digit.
For the second row, start with 6×7×2=84. Add to that 7×8×2=112, when, with two spaces reserved for this new number, you now have 8512 in this row. Then add 8×9×2=144 the same way, making this row 851344. I'll add a zero to the end to make the numbers align more clearly, making the second row 8513440.
The third row starts with 6×8×2=96. Then add 7×9×2=126, giving 9726. Add two zeroes for alignment, making this row 972600.
For the fourth row, you have 6×9×2=108 plus three zeroes for alignment, giving 108000.
Adding them up, you get 36496481 + 8513440 + 972600 + 108000 = 46090521, and you're done.
@@zanti4132 Wow ok this was very helpful.. Thank you! Can I reach out to you on instagram?
I dobt get it please help! With graphic
hey i did try the way you mentioned but i am unable to get the right answer if you are available to reply back. please do
this is literally (a+b)^2 haha
Literally. Good observation
he told that already like twice
@@tecmath To justify no idea because I have not learned algebra yet
Go to 4D space time by AK science
Yes it’s (a+b)^2
I learned this method with a enginner when I was younger, saved my life in a lot of ocasions!
Hi Josh, thanks for an easy explanation on squaring numbers up to 1000. I don't recall this type of explanation in any of my early schooling or tuition, which I shall now say it was knowledge denied. For this expanation, you deserve an A+🌟🌟🌟🌟🌟👏👍🥇.
Thank you for your like 👍. Pleasing comments and questions always appreciated.
Thanks mate
@@tecmath 👍📐Cheers.
If you keep showing these tricks, I'll never be able to leave the computer--I'll starve to death!
Whee! That was fun. I hadn't played like this since 7the grade, and that was decades ago.
Hehe 54
47 years ago for me my how the time has flown, just amazed at how old I have gotten, it was just yesterday I was learning this in math class LOL
Dude!! This is ridiculously simple...
thanks, really helped me in my 2 digit squares and the technique was unknown to me previously
This guy earned my sub, his videos are goddamn worth watching
I have a question how about d square of 696 where do u write d product if 6 and 9 doubled that will be a 3 digit number
same with 999 squared
@@Itsme7384 It actually does work for 999, you just have to contract the number. If you did this correctly, you get 162162 in the middle. What you have to do is add the two numbers in the middle to get 16362. The catch is that you have to line it up with the 818181. At the end you get 818181+163620+162 = 998001 which is 999^2. The same applies to 696.
@@Itsme7384 there's an easier method there.
The concept being used in the video is to expand it as a binomial or a trinomial and just using the appropriate number of zeros. There's another trick: use a-b for numbers close but lesser than multiple of 10, 100.
Like 696^2 = (700 - 4)^2... So 490000 + 16 - 5600 = 484416
999^2 = (1000 - 1)^2... This becomes 1000000 + 1 - 2000 = 998001
The last digit of the 3 digit number would go in the same position as the last digit when it is 2 digits. So the first 108 would go directly under 368 and the second 108 would go below the 813. That way the last digits are still under the 8 and the 3.
@@shravyaboggarapu5877 And I'm like close encounters of the third kind.
He seems to be laughing always while talking or more likely is gonna burst into laughter but somehow getting everything together
Thanks so much!!! I tried it on a 4 digit number and it worked and it basically works on any number
(a+b)^2 = a^2 + b^2 + 2ab. Nice, this is an easy way to look at it!
Looking at it as an algebraic equation just makes it easier. e.g.
Like someone mentioned (a+b)^2 or ².
Yeah exactly! (a+b)² = a² + 2ab + b².
I'd rather do it like this though
512² -> (500+12)²
= (500)² + 2(500)(12) + (12)²
= 250,000 + 12,000 + 144
= 262,144
It's kinda easier to digest. But either way works.
I agree!
What if the number is for example 998,in second step 81× 2 gives 162 ,and also 72×2 gives 144, how do you do these numbers?
Had to look at this and figure it out...it gets sloppy. You put the two numbers together but add the one's digit from 162 to the hundred's digit of 144 together getting 16344 and like standard math this goes below the first number shifting one to the left (leaving a space under the 1's digit) the other 144 in the third row is once again shifted another row leaving an empty space under the 10's digit. It might be easier to figure out than the standard squaring method, but I don't think I could do this in 5 seconds in my head.
Can you please explain this on paper 📜
You use (1000-2)² instead of the (a+b)²
what about a bigger number like 989…?
989²=(900+80+9)²=900²+80²+9²+2(900*80+80*9)+2(900*9)=816481+145440+16200=961921+16200=978121
yet another squaring method from you that caught my attention, thank you!
I decided to learn math as an adult because i have nothing better to do with my free time.... plus I regret not taking math seriously as a kid. These methods are helpful.
Me who use a calculator to calculate 1+1 : interesting
😅Bruh same
This is Vedic maths
This is simple algebra
Genius 👏🏼
Great!
With squaring the x0y numbers I'd go from right to left, just in case 2*xy is greater than 100 and you can immediately carry the one.
you solved the problem I've been having. Thanks!!!
I have no words to thank you for teaching this superb method
I homeschool my son and this guy has made it so much easier teaching him. Thanks
Helpful 👍
Me before I watched this: how does he explain a 5-second method in ten minutes -_-
Why not, we all love it
Mesmerizing tricks. This will definitely help me in my GRE quant
This was awesome!!! Will be teaching my kids your techniques
this is literally (a+b)^2 .Its amazing trick i like picking the easy way and i got one
Yeah! But it'll be easy like a²+2ab+b²!
Expansion works better!
@@lvpist 20ab bro
@@Danon60Hz noooo just 2 cuz (60+7)^2
@@sakshamsinghal5418 He has 10a and 1b, not 1a and 1b
@@Danon60Hz no 60 is a
Is there a special trick for no.s like 888 or 99, or 666,. Or 77 etc?
@@stevenbrock528 for numbers all 9s you can do 99^2= 10000-199
999^2=1,000,000-1999
9999^2=100,000,000-19999
Hey... this scales up infinitely.
Let’s test it on 62,347
Finding the squared numbers is simple...
You get:
3,604,091,649
Then you just add the numbers multiplied together:
241,224,560
36,164,200
4,828,000
840,000
To get the answer:
3,887,148,409
But at this point, you’re better off using the four square method instead.
The four square method is basically the same thing but in a chart.
Nice, what's the four square method anyway, I'm in 8th grade so i don't know about it yet
his voice made the video so much better, love the method
Finally found that man on yt that helps everybody in the last day before exam ty btw for helping me make my teacher think that I’m nuts
This is really helpful. I tried doing another one on my own. i was going for 596 squared. but i found it was tricky to do 9x6x2 because you end up with 108 which makes that line 5 digits. and im not sure how to put it in the "middle" the 60 since theres 5 digits. Thanks for your help
I saw it question and tried it and it has me puzzled as well... I did it the old way and got the answer though and I've been looking at it since an can't come up with a solution. Lets see if he replies
@@jaemielee_mc1053 thanks for your attempt. I'm sure there's an easy solution.
@@BooksWithBrad I saw another comment that explained and said to carry or add the 1 to 90 and make it 91 thus eliminating the zero.
@@jaemielee_mc1053 hmmmm okay thanks. Math is fun
90 and 108 mean in reality 90 000 and 1 080 so if you addition them, it gives 91 080 ----> 9108
what if we have 2ab more than 2 digits for example how will you solve 987 sqaured using this trick
816449
14512_
126__
-------------
974169
144___
112_
14512_ = 2*9*8*1000 + 2*8*7*10
@@didididtuiu4494 um this doesn’t make any sense? Where did the 5 come from.
816449
114112 seems like it would be correct, so please actually explain as we don’t know how you got the 5 and where the first 1 went in 114.
@@DillPicklePatrick Yes, it does make sense!
the polynomial (including the binomial one when a=0) trick in detail:
(100*a+10*b+c)^2 = 100^2*a^2 + 10^2*b^2 + c^2 +
2*100*10*a*b + 2*10*b*c +
2*100*a*c =
a^2*10000 + b^2*100 + c^2 +
2ab*1000 + 2bc*10 +
2ac*100
For the special example of 987:
9^2*10000 + 8^2*100 + 7^2 +
2*9*8*1000 + 2*8*7*10 +
2*9*7*100 =
81oooo + 64oo + 49 +
144ooo + 112o
126oo =
816449 +
14512o +
126oo =
974169
You can use the formula (a+b+c)^2
isnt this the same concept as squaring a bi- or trinomial
It helped me a lot at maths. Greetings from Brazil!
My math teacher is trash right now she just speedruns the whole thing, so thank you for this technique!!
And you make math fun too!! ✌️
hey mate, love it learnt it pretty easy but i just did good old 999^2 but you see when we start to multiply 9x9 then double it, what do we do with the 3 digits. (the ones you showed equal 2 digits which is easy)
i have the same question
Me too
Okay I found it. You just add the hundredth digit to the one it overlaps
@@teslaromans1023 please elaborate
@@abhinavdubey4029
818181
16362_
162__
-------------
998001
162___
162_
16362_ = 2*9*9*1000 + 2*9*9*10
Because I’m going to be enlisting into the navy but my numeracy is a big problem but I’m going to binge most of your videos 😂
Absolutely love your voice explaining it to us.
discovered this as a kid. and once you understand it you can do heavy multiplication using it as well.
we had a vedic math demonstration when I was in 6th class and the guy showed as the same trick however demonstration was short and he did't give much explanation.
the trick is to understand position of each multiplication by position of digit
so AB X CD : (10A+B) X (10C + D) = 100AC + 10(AD + BC) + BD i.e. (reading right to left) BD gives 1st digit with a carryover, AD + BC + carryover gives 2nd digit + new carryover , AC + the new carryover gives 3rd and 4th digit
now squaring changes simplifies the formula
also when you start practising this you realise how to use 0 to ignore some calculation and use 1 to simply write in some products
another trick in squaring ( in multiplication in general) is if squares of a 2 digit subpart ( in extension multiplication of 2 digit numbers) are known to you , you can use it to simplify it .
e.g. 512^2 take this as (500+12)^2
going right to left 12^2 = 144 i.e. 1st & 2nd digit = 44 and 1 carryover , 2 * 5 * 12 = 120
120 + carryover 1 gives 3rd and 4th digit 21 and 1 as carryover , 5^2 = 25 is 5th and 6th digite + carryover 1 = 26
so , 26 21 44
Calculator: *"Finally, a worthy opponent! This battle will be epic!"*