Expand the following powers of : = The Binomial Theorem states that where for positive integers and . Verify this formula for the coefficients you found in step 1. (Notice that the numerator in the above expression has factors.) Newton wanted to figure out a similar way to expand . , then . Here’s Newton’s idea: if Find a linear function such that other stuff of degree 2 or higher. Find a quadratic function such that other stuff of degree 3 or higher. Find a cubic function such that other stuff of degree 4 or higher. Use your answers to #3 above to create a Generalized Binomial Theorem for . How should be defined? Use your formula from #4 to find a quartic function such that other stuff of degree 5 or higher. Use your formula from #4 to write out the first 8 terms of . Newton used his theorem to approximate square roots. Use your result to #6 (and a calculator) to approximate . Compare this with your for . calculator’s approximation Newton discovered the generalized binomial theorem around 1665. Not to depress Postscript: you, but he was your Newton makes us all look bad. see age then. Don’t worry though – We’ll that generalized binomial theorem plays a key role in his attempt to approximate . By Newton’s the way, the notation was first introduced in 1826, so real calculations were a bit Newton’s messier. One last idea expansion of is the same as the Taylor series expansion, – Newton’s expanded around . for a while whether these two expansions MUST, a It’s worth pondering be the same. priori, Newton Approximates A circle of diameter 1 has a circumference of . Archimedes approximated by inscribing and 𝜋 circumscribing polygons about a circle of diameter 1. He found that an inscribed 96-gon has a perimeter greater than and a circumscribed 96-gon has a perimeter less than , thus, . For centuries other mathematicians improved on approximation by using polygons Archimedes’s with more and more sides. In 1593, Adriaen Von Roomen approximated to 15 decimal places using polygons with sides!
Expand the following powers of : = The Binomial Theorem states that where for positive integers and . Verify this formula for the coefficients you found in step 1. (Notice that the numerator in the above expression has factors.) Newton wanted to figure out a similar way to expand . Here’s Newton’s idea: if , then . Find a linear function such that other stuff of degree 2 or higher. Find a quadratic function such that other stuff of degree 3 or higher. Find a cubic function such that other stuff of degree 4 or higher. Use your answers to #3 above to create a Generalized Binomial Theorem for . How should be defined? Use your formula from #4 to find a quartic function such that other stuff of degree 5 or higher. Use your formula from #4 to write out the first 8 terms of . Newton used his theorem to approximate square roots. Use your result to #6 (and a calculator) to approximate . Compare this with your calculator’s approximation for . Postscript: Newton discovered the generalized binomial theorem around 1665. Not to depress you, but he was your age then. Don’t worry though – Newton makes us all look bad. We’ll see that Newton’s generalized binomial theorem plays a key role in his attempt to approximate . By the way, the notation was first introduced in 1826, so Newton’s real calculations were a bit messier. One last idea – Newton’s expansion of is the same as the Taylor series expansion, expanded around . It’s worth pondering for a while whether these two expansions MUST, a priori, be the same. Newton Approximates A circle of diameter 1 has a circumference of . Archimedes approximated 𝜋 by inscribing and circumscribing polygons about a circle of diameter 1. He found that an inscribed 96-gon has a perimeter greater than and a circumscribed 96-gon has a perimeter less than , thus, . For centuries other mathematicians improved on Archimedes’s approximation by using polygons with more and more sides. In 1593, Adriaen Von Roomen approximated to 15 decimal places using polygons with sides!