2. 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 and (a – b)² = a² - 2ab + b² Consider: 11² = 121 = 11x11 = (10+1)(10+1) = 10x10 + 10x1 + 1x10 +1x1 = 100 + 10 + 10 + 1 = 121 Similarly: 9² = 81 = (10-1)(10-1) = 10x10 + 10x(-)1 + 10x(-1) + (-)1x(-1) = 100 – 10 – 10 + 1 = 81 Explanation:- when we square the sum of two numbers, the result is the sum of the first numbers squared, the product of two numbers multiplied by two and the second number squared.

3. If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? we now have calculators that performs complex numerical calculations easily, but in the ‘bad old days’ it was all done with pencil and paper. So if I was an engineer 100 years ago, this method have been very much useful rather than doing long and hard multiplications in paper because its much less complicated. Its easy to workout with small squared and root numbers like 13 or 50. However for example if I wanted to find the square root or the cube root of a big number like 463 from 100 years ago, it will be much harder to find ‘463 times 463’ in paper than using a calculator and it will also take a long time to figure it out and you might not get the correct answer. Its also hard to multiply numbers containing decimals such as 0.199, 3897, 102.32, 222.2, 0.008… in long multiplications. So if this method was discovered a long time ago it will be very useful for the people and engineers to use it.

4. At what point would our method would be cumbersome? (i.e.. How many decimal places or what sorts of numbers would make us think twice about using this method?) Numbers with lots of digits will make us think twice about using Binominal Expansion method. It would be really hard to take away, multiply big numbers or very small numbers, either with a whole number or a decimal point number. Also there are lots of numbers that will be very complicated to use in binominal expansion like the numbers that are far from 10, 100, 200… like 52, 168 they are more or less the number by tens. Furthermore for example if I want to try to square 2,346,100 or 122.8866 using the long multiplication you wont need a calculator to do it because the numbers will be easy and small to multiply, however this method will take a longer time than the expansion method. Its also much easier to calculate a product by decimal numbers using the old way which you take off the decimal point …. and so on as the example below shows. (34.57) = (30+4.57)(30+4.57) = 900+137.1+137.1+20.8849 =1195.0849

5. Can you give us detailed explanations and examples of where long multiplication is more efficient than our expansion method? In my opinion long multiplication will would be more efficient than using expansion methods when multiplying by decimal numbers. Because its really hard to work out the square or the cube of a decimal number using the binomial expansion method, and you may do silly mistakes that will affect your answer like putting the decimal point in the wrong place or forgetting the zero... Furthermore its hard to multiply a decimal number for example 0.935 by any other number without using a calculator. Therefore in this case I prefer to use the normal long multiplication method which is first I add the number of digits after the decimal points from both numbers and remember it in my head. Lets say 2.667 times 1.87, so there are five digit numbers after the points, then I multiply it normally 2667x187=498729. Finally I move the invisible decimal point five backwards starting from the last number which is nine. So the answer should be 4.98729.