Numerical Estimation
“” means “approximately equal to” Recall Your Memory I 1 2 1 4 1 5 exact value = 0.5 = 0.25 = 0.2 1 3 approximate value 1 6 0.33 0.67 “” means “approximately equal to”
Recall Your Memory II Round off 8 120 correct to the nearest hundred. Round off 8 170 correct to the nearest hundred. 8120 8170 8100 8200
How many are there in our school? Case I How many are there in our school?
Case II What is the travelling time from to by ?
Case III has and would like to buy the following products in a supermarket: $101.5 $99.1 $80.7 Does have enough money?
Case IV can carry 49 passengers. Now, there are 422 and 10 to go for a picnic. How many do they need?
Is it possible? Case V would like to order 9 over the phone. Each costs $52.5. shows $5 197.5. Is it possible?
Why do we need estimation? For easier calculation When the original number is unknown The value may vary a bit in different situation Limitations in measurement Checking whether the result is reasonable
A history of (希臘) 圓周率的發展 年代 求證者 內容 古代 中國周髀算經 西方聖經 周一徑三 圓周率 = 3 元前三世紀 圓周率 = 3 元前三世紀 阿基米德 (希臘) 1. 圓面積等於分別以半圓周和徑為邊長的 矩形的面積 2. 圓面積與以直徑為長的正方形面積 之比為11:14 3. 圓的周長與直徑之比小於3 1/7 ,大於 310/71
中國 荷蘭 三世紀 劉徽 用割圓術得圓周率=3.1416稱為'徽率' 五世紀 祖沖之 1. 3.1415926<圓周率<3.1415927 2. 約率 = 22/7 3. 密率 = 355/113 1596年 魯道爾夫 荷蘭 正確計萛得p的35 位數字
法國 英國 1579年 韋達 '韋達公式'以級數無限項乘積表示p 1600年 威廉.奧托蘭特 用p/σ表示圓周率 π是希臘文圓周的第一個字母 σ是希臘文直徑的第一個字母 1655年 渥里斯 開創利用無窮級數求p的先例