Binominal Expansions

1. Binominal Expansions By: Barbara Giesteira

2. This is the binominal expansionmethod

3. Introduction (0.99)² = (1-0.01)(1-0.01) =1²-2x1x0.01+0.01² General Rule (a+b) ² =a²+2ab+b² and (a-b) ²=a²-2ab+b² So first you square the a, then you multiply the exponent by b and a, then you square b.

4. If you were an engineer 100 years ago, how this method may have been useful rather that just using long multiplication? (0.99)² = (1-0.01)(1-0.01) =1²-2x1x0.01+0.01² As you can see it is much easier to do the binominal expansions method because then you don’t have to use the calculator, which in the old days, like 100 years ago they didn’t have. If you were to do the long multiplication it would take a lot of time because multiplying decimals it is more complicated. Sothis method makes it easier because it breaks down into smaller numbers, because working with the numbers 1 and 0.01 is easier then multiplying 0.99 how many times the exponent is because the place value of the decimals are at the hundredths and the number 1 is also easy to work with because it is a small number. I think 100 years ago this method may have been helpful for engineers to use because for example, if they were constructing a building they probably would have to know how long is each side was and how wide each side was, so that they would get the proper amount of materials. So to do that first they would have to know what the height was, then they would do this method to see how long would be both sides, so you find out what the first height is, then you put the number to the power of 4 because there are 4 different sides in a building, and you use the method. Then they would have to know how long the width is. So to do that, first you would have to know how wide the building is going to be, then you put it to the power of 4, because in a buildings there are 4 sides. And from that they could find out how much material was needed. So this method was probably helpful for engineers 100 years ago because in that time they didn’t have calculator so this made it easier for them because it is easier then multiplying the same number how many times the exponent is, which the exponent may be low or high and if it is high, it would be even harder to solve. And if they were doing the long multiplication rather then the method, the calculation would get so big and complicated that they would probably get the results wrong. So without using the calculator the method it is more neat because when you do the long multiplication, your working may get so big, you will get confused and you might get something wrong. Because of the confusion you might make a little mistake and your results will be completely wrong. So 100 years ago, the method was much more useful then using the long multiplication.

5. At what point would this method be big and cumbersome? • I think this method would be very cumbersome when the exponents gets higher. Because when the power is low, like 2. This would be you working out: (0.99)² = (1-0.01)(1-0.01) =1²-2x1x0.01+0.01² So as you can see the working out is very small and easy But if this was (0.99)6 , this would be the work out: (0.99)6 = (1-0.01)(1-0.01) (1-0.01)(1-0.01) (1-0.01)(1-0.01) (1-0.01)(1-0.01) (1-0.01)(1-0.01) = 16 - 6x1x0.01+0.016 So as you can see, the working out is huge and it takes a lot of work and time when the exponent is higher. So as you can see the work gets much more cumbersome and big when the exponent is higher.

6. At what point would this method be big and cumbersome? • I think this method would also get cumbersome when the decimal place value gets to the thousandths. With the tenths and hundredths I think it is better to do the binomial expansion method because the decimal place value is smaller soit takes less time. But with the thousandths place value you can only do it if you use the calculator because it gets really complicated because there are a lot of decimal place values. So in the old days it was probably hard for them to do use the method when the number was in the thousandths decimal place because the calculators didn’t exist in that time, and it is very complicated to work with numbers with a lot of decimal place values .

7. Can you give some detailed explanations and examples of where long multiplication is more efficient than the expansion method? I think long multiplication is more efficient when the place value of decimal is at the thousandths. Because if I was to do the binominal expansion method with the number(0.999) ², this is what I would have to do: (1-0.001)(1-0.001) =1-0.001-0.001+0.000001 =0.998001 So as you can see it gets really complicated and I was only able to do it with a calculator. So when the binominal expansion method is at the thousandths, it is not efficient. So in this case it would be better to use the long multiplication. It is better do use the long multiplication when the decimal place is at the thousandths because when you do the binominal method the place values of the decimals go even further from the ten thousandths decimal place. So there would be a lot ofof the decimals place values , which are really hard to work with without the calculator. So in this case it would be much easier to do the long multiplication because in the old days they didn’t have calculators so it would be hard for the people to work with decimal place values even further then the thousandths. So if I had to choose when to do the binominal expansion method and the long multiplication without using the calculator, I would do the binominal expansion method when the number is a whole number or when the decimal place values are at the tenths or hundredths, and I would only do the long multiplication when the decimal place value is at the thousandths or further.

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