Binomial Expansions

1. Binomial Expansions By: Nawal Syeda

2. 100 years ago.. 100 years ago calculators weren’t invented yet. Calculations were done with paper and pencil, not much methods were discovered that time either. The engineers discovered long multiplication and they used it for many of the measurements taken for buildings etc, so long multiplication took place. Long multiplication wasn’t the best method they could have used. Instead the could have used Binomial Expansion.

3. Binomial Expansion Rules: • (a+b)2 = a2+2ab+b2 • (a+b) = a2-2ab+b2

4. If I was an engineer… If I was an engineer back in the old days (100 years) where you can’t solve mathematical problems by just a push of a button. Binomial Expansion would have been very useful instead of using long multiplication. With the method (Binomial Expansion) important calculations will further improve by calculating faster, more accuracy, less time will be lost and you will be able to achieve your desired result. For instance building a skyscraper applies calculations to be extremely accurate, otherwise because of the force of the air the building will collapse onto several and thousands of people destroying other properties, trees etc. This could happen because of long multiplication. Long multiplication needs time and it gets messy because you have to carry a number, borrow 1 etc. It is also easier to make more mistakes and errors while using long multiplication. This will cause to loose many lives, because of the minor mathematical errors that caused the skyscraper to collapse.

5. If I needed to build a floor of a building the length would be 20 and the width would be 20 and I had to use long multiplication my working out will end up like this: As you can see it looks messy and it is possible to make many errors because there are many zeros and you could get confused and we are humans and we are not perfect

6. We can now compare it with Binomial Expansion As you can see this is quick and clear, lesser steps and has met my obligations and it is the same answer like the long multiplication which proves my answer is correct. Long multiplication tiresome because there is a series of steps to go through.

7. Long Multiplication V.S Binomial Expansion • This concludes that Long multiplication could get very messy and once it’s messy it’s possible to make mistakes and it takes too much time 100 years ago (no calculators invented). Although Binomial Expansion takes a few seconds, its neat, organized and we have more of a chance to make lesser mistake than long multiplication because, once you use long multiplication there are many numbers and zeros to confuse. Therefore Binomial Expansion doesn’t have much zeros to confuse which causes fewer errors. This is why I believe Binomial Expansion is much more useful than long multiplication.

8. When Big and Cumbersome? • This method will be big and cumbersome at the point where we square many digits (more than 3 digits), which aren’t zeros. Also it is cumbersome when there are a lot of decimal places. This method will be useless because this causes more calculations and than long multiplication.

9. Think twice… • An Example would be when I the architect 100 years ago (no calculator) had to construct the largest building and I had to get accurate measurements or else the building could collapse and thousands of lives will be lost. It was up to me to make sure that won’t happen. The Length and with of the building was 211.02905km.

10. Examples (4376)2 = (4376) (4376) = (4000 + 300 + 70 + 6) (4000 + 300 + 70 + 6) = 4000 x 4000 + 4000 x 300 + 4000 x 70 + 4000 x 6 + 300 x 4000 + 300 x 300 + 300 x 70 + 300 x 6 + 70 x 4000 + 70 x 300 + 70 x 70 + 70 x 6 + 6 x 4000 + 6 x 300 + 6 x 70 + 6 x 6 = 19149376 (211.02905)2 = (211.02905) (211.02905) = (215.00000 - 4 + 0.02 + 0.009 + 0.00005) (215.00000 - 4 + 0.02 + 0.009 + 0.00005) = 215.00000 x 215.00000 + 215.00000 x – 4 + 215.00000 x 0.02 + 215.00000 x 0.009 + 215.00000 x 0.00005 + -4 x 215.00000 + -4 x -4 + -4 x 0.02 + -4 x 0.009 + -4 x 0.00005 + 0.02 x 215.00000 + 0.02 x -4 + 0.02 x 0.02 + 0.02 x 0.009 + 0.02 x 0.00005 + 0.009 x 215.00000 + 0.009 x -4 + 0.009 x 0.02 + 0.009 x 0.009 + 0.009 x 0.00005 + 0.00005 x 215.00000 + 0.00005 x -4 + 0.00005 x 0.02 + 0.00005 x 0.009 + 0.00005 x 0.00005 = 44533.25895 km

11. When numbers start to increase to 4 digits and when there is lots of decimal places. The method get hard because due to the increase of digits mental math will be harder to do because of complex multiplication, addition and subtraction. This method will require lots of time and it will get messy, so it is useless and you might end up with long multiplication.

12. Where long multiplication is more efficient than our expansion method? • Once our method begins to get big and cumbersome we need to start using other simpler, efficient methods, one method is long multiplication. There are some advantages using long multiplication has, like moving the decimal places so the number can be a whole number so multiplying is easier and faster. Although this method takes some time, the results are very accurate because multiplying is the basics. Binomial Expansion requires many steps like adding, multiplying huge numbers, which is tough to do it mentally. Although long multiplication applies the basics using the times table, so it’s easier. And long multiplication has only two steps multiplication and addition. Binomial expansion requires a series of steps, adding, multiplying, expanding, simplifying and sometimes even subtracting

13. Benefits of long multiplication • Long multiplication helps us find different types of measuring systems like height, speed, weight etc. We are unable to do this using Binomial Equations. We use these methods in life to find out how many chairs can fit in a room, how much water we need in pool, what time will we get to the destination we set.

14. Conclusion • As you can Binomial Expansion and Long Multiplication both have benefits and limitations. It depends on what is needed to solve and the type of question. Both methods are special in there own unique ways. Binomial Expansion could have helped the engineers 100 years ago. A time where you couldn’t solve the answer by a push of a button.