Factoring Quadratics... What If You Can't? (NancyPi)

[@nerdygiant7805]

This woman helped me get through college algebra years back. I didn't realized I hadn't subscribed until you popped up with this video. Nancy, please keep popping out more videos. I might not be taking the class anymore but I still enjoy the content.

[@josemath6828]

Muy bien.

[@gamergeto6437]

Bc kisak chutia bna r

[@Bhargav_Studies]

Is she a teacher or something.....????

[@-_IT_-]

Same!! A life saver.

[@daithi1966]

I hope we don't have to wait for another year to get another video. Nancy is easily one of my favorite Math KZbinrs, and over half a million people apparently agree.

[@eternaldoorman5228]

She is really special. She does it all in mirror-writing too!

[@kiyotakaayanokoji3065]

@@eternaldoorman5228 I dare her to wear shirt with text 😅

[@eternaldoorman5228]

@@kiyotakaayanokoji3065 Shhh! She fooled me! 😂

[@daithi1966]

@@eternaldoorman5228 She's also right handed, which makes it even more impressive.

[@eternaldoorman5228]

@@daithi1966 😂

[@AmberWolf353]

I don't even need to learn about factoring quadratics. I was just happy to see you posted a new video! You were a life saver during my college algebra classes. Thanks for all you do and welcome back : )

[@FMFvideos]

please come back! numbers doesn't make any sense without you

[@undercoveragent9889]

lol That made me giggle.

[@perhapsmaybe119]

You're the undercover teacher!! 📢​@@undercoveragent9889

[@russmonte5176]

Thankyou nancy.i lost all my math from a stoke.when I was in my hospital bed I couldn't figure out how many hours from 8pm to midnight .now your helping with factoring quadradics and basic trig.i forgot what foil was till you mentioned it

[@MrU4theChillWind]

Welcome back. Not a mathematician, but I love math and you give great quick explanations.

[@josemath6828]

Muy bien.

[@matildamoonga7775]

Omg Nancy! Welcome back 💕. You've been awfully missed. Seriously hate math a little less thanks to your extremely helpful videos.

[@SirWolf2018]

OMG she's really back! Can't believe this, I'm so glad!

[@rain_jen257]

Yeahh mee too!!! soo happyy!!!

[@immortal4942]

me too I can't believe it

[@neilmclachlan3931]

Yay.

[@josemath6828]

Correcto.

[@sonasunny5423]

You don't know how much you are helping us. I literally cleared my maths papers only because of you! You literally saved me in highschool and now in college. Thanks for everything.

[@plat216]

hol up she posts college level stuff??

[@00ANURAG15]

@@plat216 😆😂Lol

[@ampsquadfanclub3662]

🧢

[@josemath6828]

Así es.

[@Aspirant19131]

Cap

[@elevolucionestoica]

This class is a masterpiece! Your teaching style is so enthralling that it sparks curiosity and fuels a deep desire to learn more.

[@DRYusuftheGreat]

My teacher, I will be always grateful for you, you got me through so much in high school. from a 1st year student of medicine.

[@CarlosMartinez-jo8vl]

Nancy’s return just made my day, thanks a lot for the content and we really hope you continue making these awesome videos

[@josemath6828]

Awesome thanks

[@alphagil]

Welcome back!

[@11Kenjii]

Komi can't communicate 😳

[@Moemoney556]

Yes

[@tokyodami2635]

Anime goes stupid

[@Moemoney556]

@@tokyodami2635 really wondering who tf asked

[@AlexCraxkerShitposting]

@@Moemoney556 Ñ

[@tkcphoenix2k4]

My kids were just working on homework that had a quadratic that couldn’t be factored and we couldn’t figure it out… this video saved our sanity.

[@joshuaspector8182]

Did u solve it?

[@pinpon163]

And your pride lol

[@ayushrajsingh9652]

😂😂

[@Grassmpl]

You can also use the rational root theorem to test the quadratics.

[@rykazou183]

You’re a bit lucky this is only video she’s made in a year 😅

[@tyblue1229]

Nanc, you make math understandable, which my professors fail at. I would LOVE to see you continue on with math videos (specifically calculus 1-3). I know it takes time and work to make these videos so I am prepared to buy you all the pasta, just please come back!

[@arkprince9413]

this channel 4 years ago college first calculus 1 and 2 helped me a lot I recommended it to everyone back then and everyone liked your videos

[@vincentortega4284]

Welcome back Nancy, you have helped me so much with your awesome teaching. Glad you're back Nancy.

[@josemath6828]

Awesome.

[@wanderer3723]

Welcome back Nancy!!! Thank you so much for your great videos - they have been really helpful for quick revision before examinations! I hope to see more such amazing videos soon - you surely are the best KZbinr and I'm sure millions of people across the globe agree!

[@dylancagle885]

So glad you posted! A good math lesson is just what I needed

[@renedavids6148]

Came across ur 1st video to help my daughter. It was super super easy to understand. She came home from her test and the first thing she said was I did the magic x and nailed it. Thank you!!!!!

[@SortaDopeIGuess]

Thank you! This has been so helpful to my nephew whose struggling with his hs school teachers that aren’t really good at explaining things logically. So Please don’t ever stop!

[@Mandark020]

I've got a degree in mathematics already, but for some inexplicable reason I just can't resist the urge to watch this video on factoring quadratics.

[@indogoUI]

😂

[@petergianakopoulos4926]

Bc she is a 7?

[@fredettaboutit]

@@petergianakopoulos4926 alittle harsh , I’d give her an 8

[@petergianakopoulos4926]

@@fredettaboutit I was being a jerk on purpose.. definitely 8

[@dunemae]

Stop rating people you're gross. Especially when you can't compare.

[@CiedJacaba]

Two things I learned from this video: 1. some Quadratic Equations could be unfactorable 2. Nancy is human

[@billjamal4764]

Nah, no way. "Im human too" is exactly something someone who isn't human would say

[@1Konu1Zoru]

2:20 ALL QUADRATICS CAN BE FACTORED.. just use DISCRIMINANT and there you go.. you can factorize into two she looks so professional.. and she can edit those parts but instead she said: i am a human i made a mistake..etc..etc... but i hoped, she first says this then she make the factorization this way: D=16^2-4.1.51=52 {x-[16+sqrt(52)]/2}.{x-[16-sqrt(52)]/2}

[@1Konu1Zoru]

that would be LEGENDARY

[@MohitBaboria]

​​@@1Konu1Zorusome quadratic equations cannot be factored those who have a negative discriminant however they probably don't have any solution in the real numbers

[@wayne9287]

@@billjamal4764 Tell me what is it to be human? What is a human? Because to this day no one has given me an answer.

[@beteb4626]

So happy you’re back! I passed freshman Calc with an A- in 2015. I had a terrible professor, but your videos taught me enough to do well in the exams. Thank you and welcome back👏🏽 Would love to see Statistics related videos🙏🏽

[@josemath6828]

Correcto.

[@josemath6828]

Thanks

[@abhinandanprakash7027]

Why would KZbin recommend me this video 🤔!!

[@zentyler1646]

You took Calc in freshman year?

[@keb990]

She helped me get my bachelors in Aeronautics. These videos allowed me to understand important key concepts. Thank you!

[@thealienobserver9114]

I just wanted to pop in here years later and say, thank you NancyPi. Your videos got me through college pre-calculus about 4 years ago, and now I'm on my way to becoming a dietician.

[@nouraldinalali6290]

I would love to see some harder topics from calculus 2 or 3, you explain so well and it might help alot of people because nobody understands their professors.

[@JakesMathLessons]

I have a bunch of these on my channel if you want to check them out. I hope they help! Let me know what you think.

[@lawrencejelsma8118]

​@@JakesMathLessons... Actually when we are interested in the imaginary numbers plane (ax + jb)(cx + jc) where j is defined as ✓(-1) if we aren't visualizing complex numbers on the polar number plane consisting of an imaginary axis j and a real axis x then the best solutions are from the quadratic equation ax^2 + bx + c factors into a (x/(2a) + 1/(2a) . ✓(b^2 - 4ac)) (x/(2a) - 1/(2a) . ✓(b^2 - 4ac)) so we don't say all the time there are no solutions to ax^2 + bx + c. There are factored solutions on a real axis and imaginary axis or plane defining those solutions. She then has to replace the words with "no solutions" to "no real axis solutions" of the plane actually factoring ax^2 + bx + c. Going further into calculus we consider factors as polar imaginary axis and real axis solutions of all polynomials (not just quadratic) making use of trigonometry describing the solutions.

[@JakesMathLessons]

@@lawrencejelsma8118 yeah that’s exactly right! You can always find real or imaginary roots of a quadratic function using the quadratic equation. You must’ve watched my video about that, good work!

[@lawrencejelsma8118]

@@JakesMathLessons ... I learned from Mary L. Boas Department of Physics of DePaul University in Chicago, IL coursework of "Mathematical Methods in the Physical Sciences." Actually I'm a graduate student in Electrical Engineering that figures out all larger than power 2 (or maybe 3) polynomials now forever by magnitude and phase calculations (or polar visualization); where, Z = M |_cos ∆ + j M |_ sin ∆ where cos (j∆) = {e ^( j ∆ ) + e^( -j ∆ )}\2 and sin (j∆) = {e ^ ( -j∆ ) - e ^ (j ∆)}\( 2j) for "Electronic Filters or Electric Circuits Theory" coursework on "Frequency Charts" analysis (Low Pass Filters, High Pass Filters, Bandpass Filters, Butterworth Filters, Chebyshev Filters, etc. designs). It turns after Ax + B polynomials back to A'x^n + B'x^(n - 1) + ... + K° polynomials back to nonlinear trigonometric defining linear complex numbers using Matrices for solutions again.

[@Lynxxezz]

I really hope you come back. You just make learning math miraculous. You're really funny and lively. I wish you could do stuff like cubic functions and calculus. We hope you're doing well. We all miss you NancyPi 😢

[@jennifervercezes]

Nancy, idk if you may still come back to the channel, but I came to say: Love your work here, as a physics major I'd like to see the continuation of calculus from you. ❤

[@chevonnagrubb223]

❤❤❤

[@chevonnagrubb223]

WE WILL HAVE TO GO BACK AND GET THE TICKETS 🎟🙌

[@chevonnagrubb223]

I AM GOING TO BE OUT THIS WEEK FOR WORK AND WILL NEED A FEW THINGS FOR ME AND MY PARENTS AND MY FRIENDS AND FAMILY AND FRIENDS AND FAMILY

[@mirsolihmirsagatov6593]

Hi @jennifervercezes, watching your playlists and your channel description I thought we have things in common to hang-out a little bit 😊

[@OldTomato44]

Nancy's back!!!!! Nancy you are literally the only reason I passed my college Calculus classes, I want to express my sincerest gratitude for your clear explanations of difficult concepts. Thanks for your help and hard work, and I'm glad you're back! The world needs more people like you.

[@actuary33]

She can factor me all day! Vavoom!!

[@Lemuis]

The algorithm brought me here. Your math didactic, your marker wizardry, your soothing confident voice…I’m crushing hard. 🤩😍 Great channel!

[@khadijahabibi4979]

W rizzzzz

[@thebravestkid]

Just wanna let you know I’ve done my maths paper yesterday. Big thanks to you Nancy 😁

[@mikeo.4203]

Can it be! It is! She’s back with another video!!

[@kaibelzer2312]

i remember all of you’re videos when cramming for a algebra 2 placement test, glad you’re back

[@josemath6828]

Thanks

[@travisf912]

Welcome back! I hope you’ve been well. You got me through college. Thank you so much!

[@tronglieu6449]

You are one of the greatest Math KZbinrs on Earth. Millions like to you and subscribed. Thank you.

[@valdezfam]

Welcome back! Love these videos. I get Richard Feynman type vibes. Simplifying complex things so gracefully is remarkable to watch. Thank you

[@josemath6828]

Thanks.

[@bbiss3312]

Helpful tip: If you want to determine if the polynomial is factorable, find what the discrimnant is. The discrimant formula is:sqrt(b^2-4ac) and if the result of this comes out to be a perfect square, then it is factorable. If not, then it is prime (not factorable). (another helpful tip: polynomials are in the form : ax^2+bx+c)

[@Farouk.khettab]

You don't really need D to be a perfect square. When b²-4ac > 0 there are two solutions (x1 and x2 where x1,2 = -b (- or +) sqrt(D))/2a ) then the quadratic formula can be factored as following a[(x-x1)(x-x2)] When b²-4ac = 0 there is one solution (x1 = -b/2a) then the quadratic formula can be factored as (x-x1)² (e.g. x²-2x+1) when b²-4ac < 0 there is no solution (i.e d

[@theodorefreeman]

@@Farouk.khettab The example in the video "x^2 - 16x +51" fails using b^2-4ac > 0: -16^2 = 254, 4*1*51 = 204, 254-204 = 52, yet the problem is not factorable.

[@tipithedhedhistar4132]

@@Farouk.khettab because of The fundamental theorem of algebra all polynomials can be factored, and yes you could factor the one in the question, but you can’t factories it using the method that is given unless your brain is just that amazing 😂.

[@pablomalaga4676]

@@tipithedhedhistar4132 I don´t know all this BS of not being factorable: (x-11.61)(x-4.394) are the factors. They aren´t pretty like (x-2) or (x-3) but are factors anyway

[@hiimluck3r]

@@pablomalaga4676 holy, i finally found this section of comments. Yes, real numbers still can be factors, so why do we say that sqrt(13) makes our equation not factorable?

[@jaiberdiaz190]

Hey Nancy!! I missed your presence, I'm glad to learn with your videos 🥰

[@josemath6828]

Thanks.

[@zedlexslife]

I love Nancy, litteraly I love her, watching her videos gives me a sense of satisfaction

[@mad13ee]

Still my favorite math tutor ever. Glad to see a new video. Hope you're well and life is treating you good.

[@logicawe]

I passed calculus, but I'm down to learn some more when Nancy is teaching.

[@a.f1448]

😂👍🏼

[@indiegamedevelopment]

Yes I love her 💔💔💔💔

[@yansantiagosiregar8927]

heyyy theree Nancy. I'm a student in Indonesia, and I'm facing AS level examination right now, and btw, I LOVE your videos💗💗💗, and it honestly helped me through my igcse times, and I'm so grateful for it. why haven't you made any videos lately? I'm really hoping to see more insightful videos especially for us students that is going through examination right now. THANK YOUUU SO MUCHH🙏🙏

[@khloudahmed7554]

I’m so glad you updated!! Thank you for the lesson

[@josemath6828]

Thanks

[@adrianajimenez4543]

Thanks for coming back!! Just in time for saving my maths class! Ly tons queen 🥰🥰💕💕💕

[@stocckz-5038]

just wanna thank you , you helped me get a full mark on an exam 4 years ago

[@lucc2917]

you can always use the abc formula " x=[−b±√(b2−4ac)]/2a. " where the a is for the number behind x^2 b for numbers behind x and c for the random number at the end ax^2+bx+c = 0 example: x^2-16x+51 = 0 here a = 1 b=-16 c=51 and when we put them in the formula we get a x1=11.6 and x2=4.39 this also means the factoring quadratics is the following: (x-11.6)*(x-4.39)

[@pulpogigante]

I believe this is correct 💯 Quadratic formula!

[@shiraziahmedsheikh6365]

A video after so long time. I really love to learn through your videos. You help me pass my calcus. Where were you?

[@benjaminp4369]

Con estas Maestra si dan ganas de estudiar lo que sea ,necesitamos más Maestras así 😍😍😍

[@hentali]

No matter how difficult, these questions entered my mind at the speed of light because of you

[@okami1331]

Wanted to say thank you and appreciate your work I just received my associate and watching your videos helped me achieve it, along with many The Office rewatches. Thank you N.P

[@gregsavitt7176]

For anyone who is worried, factoring becomes easier and easier the more you use it! Don’t be too worried if it is challenging at first. Its like learning a language. Practice makes perfect!

[@MrIkesimba]

10/10, the math was interesting too.

[@celsio2230]

Actually, both of those can be quite easily factored, just not with integer numbers. Just use the quadratic formula and find the values that nullify the polynom. For example, for the first one those values are 8 + √13 and 8 - √13 so it can be factored into (x - (8 + √13))(x - (8 - √13)) Edit: Sorry I don't remember the name of the theorem that defines that but basically if a value a nullifies a certain polynom then it is divisable by x - a

[@sniperwolf50]

I think you're talking about the remainder theorem. It might also be the fundamental theorem of algebra

[@celsio2230]

@@sniperwolf50 Yeah it's a consequence of the remainder theorem. Thanks!!

[@williamperez-hernandez3968]

For the second example, first factor the 3. Use quadratic equation to find the two roots, say x = e, f. Then the factorization is 3(x-e)(x-f). Finally, distribute the 3 into either parenthesis. So answer is (3x - 3e)(x -f) or (x -e)( 3x-3f).

[@702Chap]

She does state in the descrption "Usually, a quadratic expression can be factored, but some cannot be, at least not with integer numbers."

[@josemath6828]

Correcto.

[@IdkIdk-uo6ls]

For some reason I pay more attention her videos than any other teacher on KZbin

[@cthu1601]

Ohh, u're backkk! I've made several robots thanks to your insight in maths. Really appreciated and looking forward for more. Thanks a millionnnnn.

[@andreimiga8101]

Well if it just says "factor" then there's no assumption that you neee to use integers. The first one is x²-16x+51=x²-16x+64-13=(x-8)²-sqrt(13)²=(x-8-sqrt(13))(x-8+sqrt(13)). The second one: 3x²+4x-16=1/3(9x²+12x-48)=1/3(9x²+12x+4-52)=1/3((3x+2)²-(2sqrt(13))²)=1/3(3x+2-2sqrt(13))(3x+2+2sqrt(13))

[@spacecomunication]

exactly lmao, and think that people in comments thank her for those kind of maths she preaches lol

[@philipberggren1823]

Indeed

[@diegoaragon6297]

The signs of your factors are wrong... It's (x+8+(13)^(1/2))(x+8-(13)^(1/2)).

[@andreimiga8101]

@@diegoaragon6297 (x+8+sqrt(13))(x+8-sqrt(13))=(x+8)²-sqrt(13)²=x²+16x+64-13=x²+16x+51, but the original polynomial is x²-16x+51.

[@he11okittyfan]

Where did this diva go❤

[@hisoyuno]

LMAO HELP

[@LuminaryAluminum]

Speaking as a human myself, I find nothing suspicious at all about how often and how profusely Nancy points out that she is, in fact, human. I am also very human and I constantly talk about being a human and all the human things that come with the human experience, like having warm blood, eating food, and sleeping, haha.

[@josemath6828]

Awesome.

[@deniswastaken]

I feel bored while learning maths and things go off the top of my head but recently I was looking for mathematics channels available on YT and found this channel, although it's not something what I have to learn ( a bit complex than this ) but still it can be very helpful to brush up the basics. I saw that creator stopped posting 2 years ago, I hope that she is still teaching to those who need it, she surely knows how to teach well.

[@Eren-gq8xu]

I don't even know what you are talking about, I'm just literally staring at you because your voice hypnotized me.

[@comoyun]

can't concentrate

[@vanshgourkar1100]

Bro I see you had commented this 7 mo ago I just wanted to tell you that it's been 7 mo and she hasn't posted yet 😭

[@hindustani8621]

​@@vanshgourkar1100na it's been a year

[@VladimirPutin-nk6vc]

😂😂😂😂😂

[@hindustani8621]

@@VladimirPutin-nk6vc hlo vitamin protein

[@VladimirPutin-nk6vc]

@@hindustani8621 shhhhhhhh

[@silhouetteshadow18]

Hi Nancy! Welcome back! Your videos have been so helpful, I have been watching them since I was in Algebra 2. Now I am in Calculus, and I was wondering if you could cover more calculus 1 topics, specifically optimization, Newton’s method, Mean Value Theorem, etc. These calculus topics are seriously getting tricky! But nothing that we can’t handle 😊😊😊

[@josemath6828]

Thanks.

[@JakesMathLessons]

I have a bunch of these on my channel if you want to check them out. I hope they help! Let me know what you think.

[@nocturnalvisionmusic]

Awesome username Silhouette. 🥰

[@dougmckisson2873]

Just wondering/asking for a friend, exactly how tall are you?

[@babbyyobama1729]

I am the friend

[@blaintonrau5960]

I’m the friend

[@corbinmacrae2752]

I'm the friend

[@kailash_Venkatanarasimhan]

We are glad having you back Nancy:))

[@andreduplooy5731]

Ah welcome back Nancy!!! Really enjoy your content and I frequently recommend your channel to my students!!!

[@Mrmathias-qi7yv]

I think the reason she stopped posting is Because she probably started a family or has something more important. Great channel though! Keep posting even if you haven’t posted for 2 years

[@siddhantsahu4529]

we are waiting for your comeback pls come soon 🙄 pls guys like this so maybe she will see it

[@gabrielcanlas4458]

NANCY WHERE DID YOU GOOO?!?

[@johnstuder847]

Miss your new videos! Could you do a series on Linear Algebra? It’s normally very dry, but I love how you explain why we do things, not just how.

[@mattyg704]

Thank you so much for these videos. I am a returning student that needed a crash course on so many topics that you cover on your channel. I really appreciate your style of teaching.

[@zayedmulla]

Helloo Nancy ma'am, I am finding difficulty in solving integration problems, so can u please upload video on integration, that would be a big help, I just now found this channel and I loved the way u are teaching Thank you....🙂🙂🙂🙂

[@josemath6828]

Awesome.

[@roroo]

kzbin.info/aero/PL3j1ntBPCU_oXcHmmyCsSI3HcyKFj8y4E

[@mauroromel6692]

Deveria ter um tradutor como o canal está disponível aqui, deveria ter esse ou essa tradutora. Cálculo muito interessante.

[@aadeshshorts]

SO WE CAN ALL AGREE THAT THIS NEVER DISAPPOINTES US WITH HIS VIDEOS 💕

[@Helpadoggoreachsubs]

Shut up bot

[@alirezasadeghifar3815]

Nancy is the best, I always tell my students to watch her videos!!! I hope to see her soon again back in the science education scene

[@Risk-on1]

4k Nancy is amazing. we love it, please do more!

[@sentient1640]

WHY DID YOU STOP MAKING VIDEOS? 😭😭

[@6ayzid]

Is she coming back 😭

[@StaticBlaster]

I hope so. She's a great explainer always expounding on complex math topics.

[@LathaLatha-si6ti]

With that the angel got tired and she is off hibernating....... SHE WILL RISE AGAIN! ShE hAs To........

[@Elisha-phiri]

You really helped me big time with your tutorials to clear my 100 series courses in my first year.You're amazing.

[@davidgolf3245]

If I had a teacher that looked like you, I would have understood math much better. Take care and happy holidays.

[@elmismisimomaldito]

nancypi where are you?😢

[@KeabetsweDiratsagae]

Where are you Nancy😭

[@simranjeetsingh5632]

AVENGERS ASSEMBLE 😂

[@fountain9886]

woooo, you are the best tutor i have ever come across in the math.

[@comedycontent867]

Nancy I surprised to see 🙈 🙈 you back again 🥺😭 Thanks for coming 😭😭🙏🙏

[@aurora.s_dream]

come back nancy :(

[@waltuh2608]

turn back chef! army needs your help.

[@harshitparihar2497]

Please upload new video Nancy mam Please Please Please Please Please

[@saritabajpai6983]

Welcome.... Was waiting for a long time for your videos... Pls keep posting

[@alsafialialsafi6513]

Thank you very much, teacher. I am an Arab and I succeeded in the external exams because of you. Thank you very much.

[@AYMAN-P2]

عراقي

[@alsafialialsafi6513]

@@AYMAN-P2 No, I am from Saudi Arabia. Are you Iraqi?

[@AYMAN-P2]

@@alsafialialsafi6513 أي اني عراقي ارحب مليون

[@AYMAN-P2]

@@alsafialialsafi6513 Yes, I am Iraqi. I welcome you a million times.

[@alsafialialsafi6513]

@@AYMAN-P2 Thank you, brother. We are the only Arabs who spoke to me in Arabic.

[@DM-rb2qt]

I'm here as a man of culture who loves math as well.

[@Sureboiyo]

We all know why you're watching...

[@AndyJxmes]

Real

[@_DusH_]

Anyone 2024?

[@neurospizz75]

Good to see you back. Math was hard when you were gone

[@noggodtv]

This woman should have millions of subs! She’s amazing

[@Mandq.]

anddd she left us

[@wushan8487]

I really don't care about what your doing , I clicked because your super cute and I'm very attracted to you , merry Christmas from London x

[@Theguywithspectacles]

Wait she disappeared

[@TheJanvicgwaps]

What impresses me the most is how she writes every thing inreverse so its right side up for ua viewers.. So cool.

[@gregrich1964]

Hello NancyPi. I hope you know how much you are appreciated. Thanks for sharing your gift as a wonderful teacher. You gave me confidence in math.