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Introduction

Introduction. From the investigation where we looked at (0.99) 2 = (1-0.01) (1-0.01 ) = 1 2 – 2x1x0.01+0.01 2

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Introduction

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  1. Introduction • From the investigation where we looked at (0.99)2 = (1-0.01) (1-0.01) = 12 – 2x1x0.01+0.012 • We came up with the general rule for expanding binomials, in particular squaring the sum and difference of two terms: (a+b)2 = a2 + 2ab + b2

  2. Binomial Expansion • 100 years ago, there were no calculators, or easy methods of calculating, this made it very hard for people such as engineers to calculate large quantities or measurements, for example, an engineer could calculate the area of a large square piece of land by substituting a and b for numerical values: • (a+b)2 = a2 + 2ab + b2 would become (100+1)2 = 1002 + 2(100)(1) +12. This would then conclude to the result: 10,201m2 . Binomial Expansion is a good method to use instead of long multiplication but, at a certain stage it can get just as complicated. When working with large number and decimals, Binomial Expansion can get very messy, very fast.

  3. Indices • Binomial Expansion can also get cumbersome when working with different powers, eg. cubed(3) etc.

  4. Cumbersome? Very! • Binomial Expansion works better than long multiplication only to a certain extent. When the numbers start getting larger, (ie. More decimal places or not easily rounded) then the squaring method gets harder to work with. For example, if I were to square the number 973.003, it would look like this: • (973+0.003)2 = 9732 + 0.0032 + 2(973)(0.003) = 946,729 + 0.000009 + 5.838 = 946,729.838009

  5. This equation, which would have taken seconds on a calculator, took me 15 minuets by hand because I had to work out the long multiplication using a binomial structure. As you can see this number is very large, but the number that was being squared wasn’t, by simply adding a few decimal places and making the number into one that is not easily broken down, (not easily rounded) the equation and method gets very complicated. Therefore, I would think twice about using this method when the numbers you are working with start getting harder to multiply, round or breakdown in your head. This is another example where this method can get very cumbersome: • 7772 = (700+77) = 7002 + 772 + 2(700)(77) = 490000 + 5929 + 107800 = 603729

  6. Large numbers, although easy at first sight, can get larger and larger, and before you know it, your working the same thing over and over again because it doesn’t look right, and it’s getting confusing.

  7. Long Multiplication or Binomial Expansion??? • Long multiplication come in handy at times when binomial expansion gets too cumbersome, this is because even though it takes some time and space, you are able to see every step that needed to be taken to get the answer. If you then find that the answer in incorrect you can go back and trace your steps, whereas when using Binomial expansion method, we tend to skip some steps without realizing, this makes it hard to check your answers.

  8. When the numbers start to get large and cumbersome to work with, we write the answer down using the binomial method, but on the side also write down some short multiplication, this just adds up to doing twice the required amount. For example if I was to work out 1235.462 using the binomial method it would look like this: • (1235+0.46)2 = 12352 + 0.462 + 2(1235)(0.46) = 1525225 + 0.2116 + 1136 = 1526361.412 – rounded to 3 decimal places. • This does not include the multiplication done on the side.

  9. Whereas using long multiplication it would look like this:

  10. To Conclude…… • Binomial expansion would have helped a lot of engineers back in the bad old day but, only to a certain extent. Long multiplication allows you to see every mistake, or step you have made. They both have their good and bad sides, you just have to know when to use them.

  11. The End • By Arshia Jain 8C

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