# Multiply Two Numbers Near 100 Faster Than Your Calculator ## Метаданные - **Канал:** polymathematic - **YouTube:** https://www.youtube.com/watch?v=mXejbsXUQYA - **Дата:** 11.05.2026 - **Длительность:** 3:00 - **Просмотры:** 2,497 ## Описание If you like the math in today's video, I'd love for you to check out my new number sense app! Build up your number sense a day at a time: https://studio.com/apps/polymathematic/numbersense. What's 108 × 103? It turns out you can answer that in your head in about two seconds, and the method works for any two numbers hovering near 100. Start with 108 × 103. Add the "extras" past 100: 8 + 3 = 11. Multiply the extras: 8 × 3 = 24. Stick them together and you get 11,124 — which happens to be exactly 108 × 103. Try it with 109 × 105. Add the extras (9 + 5 = 14), multiply them (9 × 5 = 45), and you get 11,445. This is one of those tricks that feels like a magic act the first time you see it, but it's actually a small window into how multiplication really behaves around a "nice" anchor number. The anchor here is 100, and once you see why 100 is doing the heavy lifting, the trick stops being a trick. When you write 108 as (100 + 8) and 103 as (100 + 3), and you multiply them out the way you'd multiply binomials — partial products, or FOIL, or the box method, whichever your teacher called it — you get four pieces: 100 × 100, 100 × 3, 100 × 8, and 8 × 3. That's 10,000 + 300 + 800 + 24. Notice what happens next: the 24 has no friends in the ones or tens place, so it lands at the end of the answer untouched. The 300 and 800 combine into 1,100 hundreds, which together with the 10,000 makes 11,100. Drop the 24 on the end and you're at 11,124. The "add the extras, multiply the extras" rule is just the partial product layout collapsed into something you can do mentally. What about numbers a little smaller than 100, like 94 × 97? Same idea, just from the other side. Now 94 is 6 below 100 and 97 is 3 below 100. Multiply the distances: 6 × 3 = 18. That's the last two digits. The first part is going to be a bit less than 10,000, because both numbers came in under 100. How much less? Add the distances (6 + 3 = 9) and subtract from 100 to get 91. Tack on the 18 and you have 9,118. Calculator agrees. If you write 94 as (100 − 6) and 97 as (100 − 3), the same partial product structure explains it. You get 10,000 − 300 − 600 + 18. The two negative pieces total −900, knocking 10,000 down to 9,100, and the +18 from the two negatives multiplying nudges it up to 9,118. The signs flip, but the architecture is identical. That's the part worth lingering on. The reason this works has nothing to do with 100 being special on its own. It's that 100 × 100 = 10,000 is a clean anchor, and the leftover pieces in (100 ± a)(100 ± b) are small enough to slot into the answer without crowding each other. It's the same math that runs underneath multiplying any two binomials, and it's the same math that will show up later when students start factoring quadratics. A good mental math trick is one that's easy to use and also points to something real underneath. This one earns it on both counts. #mentalmath #multiplicationtrick #numbersense Watch more Math Videos: Math Minis: https://www.youtube.com/playlist?list=PLrc8spN1b3jkQynJ5heNvSs72gCPj_hwj Math Minutes: https://www.youtube.com/playlist?list=PLrc8spN1b3jmVFYwHiuMzCYJ0y_yiwUge Number Sense: https://www.youtube.com/playlist?list=PLrc8spN1b3jksKkY_oarNFEFAHjrIV97U MATHCOUNTS: https://www.youtube.com/playlist?list=PLrc8spN1b3jlqDkZXtby9lnEaOBy7-ZY0 Follow Tim Ricchuiti: TikTok: https://www.tiktok.com/@polymathematic Mathstodon: https://mathstodon.xyz/@polymathematic Instagram: https://www.instagram.com/polymathematicnet Reddit: https://www.reddit.com/user/polymath-matic Facebook: https://www.facebook.com/polymathematic ## Содержание ### [0:00](https://www.youtube.com/watch?v=mXejbsXUQYA) Segment 1 (00:00 - 03:00) Maybe they taught you this in school, maybe they didn't. But I'm going to write down 8 + 3, which makes 11. And 8 * 3, which makes 24. And 108 * 103 is indeed 11,124. I'm not deceiving you. This isn't a weird trick. This will keep working. 109 * 105. 9 + 5 makes 14. 9 * 5 makes 45. And 109 * 105 is 11,445. Okay, but what if they're slightly smaller than 100? Well, we're going to adjust a little bit. We don't want to think about 4 and 7 now. Instead, we want to think about how far away are 94 and 97 from 100. 94 is 6 away. 97 is 3 away. And 6 * 3 is 18. That's going to be the last two digits of our product. But what about the first bit? The numbers that were a little bit bigger than 100 gave us back products 10,000 because 100 * 100 already is 10,000. But these numbers are a little smaller than 100. And so our answer is going to need to be a little smaller than 10,000. How much smaller is going to come from subtracting something. These numbers were 6 and 3 away from 100. 6 + 3 makes 9. And then subtract that from 100, which makes 91. And indeed, if you went to a calculator, 94 * 97 is 9,118. You can see why this isn't even really a trick. If you use something called partial products, imagine your 108 not as 108, but instead as 100 + 8. And similarly, your 103 is 100 + 3. What we're really multiplying here is 100 * 100, which makes 10,000. 100 * 3, which makes 300. 100 * 8, which makes 800. And then finally 8 * 3, which makes 24. You'll notice as we add these all up, the 24 just stays 24. There's nothing in the tens or ones place in any other number here to change it. And so I see it all the way through to the final product. Similarly, there are no hundreds to add together other than the 300 of the 100 * 3 and the 800 8. And so the total number of hundreds ends up being,00. The beautiful thing about this method is it's not limited to only thinking of the numbers additively. I can just as easily think of 94 as 100 minus 6 and 97 as 100 minus 3. And when I do that, I still have 100 * 100 makes 10,000. 100 * -3 makes -300. 100 * -6 makes -600. And then finally, -3 * -6 is pos8. A good math trick to me is one that is easy to apply, but also hints at some kind of underlying structure. The structure here is one that students can take advantage of all the way through when they start multiplying out binomials and factoring quadratics. If you like this kind of thing and want to build more Numbers Sense intuition, I would love it if you would check out the link in description. I've released a numbersense coaching app and I would love for you to check it --- *Источник: https://ekstraktznaniy.ru/video/51305*