0の0乗

Transcript

1. Evaluating the 0th Power of 0 Natsu Ozawa
2. Zero to the Zeroth Power 03 = 0 × 0 × 0 = 0 02 = 0 × 0 = 0 01 = 0 00 = ?
3. Zeroth Powers 23 = 2 × 2 × 2 = 8 22 = 2 × 2 = 4 21 = 2 20 = 1 ÷ 2 ÷ 2 ÷ 2 33 = 3 × 3 × 3 = 27 32 = 3 × 3 = 9 31 = 3 30 = 1 ÷ 3 ÷ 3 ÷ 3 𝑛𝑛3 = 𝑛𝑛 × 𝑛𝑛 × 𝑛𝑛 𝑛𝑛2 = 𝑛𝑛 × 𝑛𝑛 𝑛𝑛1 = 𝑛𝑛 𝑛𝑛0 = 1 ÷ 𝑛𝑛 ÷ 𝑛𝑛 ÷ 𝑛𝑛
4. X to the Xth Power 11 = 1 0.50.5 = 0.707 … 0.30.3 = 0.696 … 0.10.1 = 0.794 … 0.010.01 = 0.954 … 0.0010.001 = 0.993 … 𝑦𝑦 = 𝑥𝑥𝑥𝑥
5. Taking the limit of 𝑥𝑥𝑥𝑥 Limits: calculate what the value would be at a point arbitrarily close to 𝑥𝑥. lim 𝑥𝑥→0+ 𝑥𝑥𝑥𝑥 “The limit of x to the xth power as x approaches 0 plus.”
6. Solution to lim 𝑥𝑥→0+ 𝑥𝑥𝑥𝑥 11 = 1 0.50.5 = 0.707 … 0.30.3 = 0.696 … 0.10.1 = 0.794 … 0.010.01 = 0.954 … 0.0010.001 = 0.993 … lim 𝑥𝑥→0+ 𝑥𝑥𝑥𝑥 = 1 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = lim 𝑥𝑥→0 𝑒𝑒ln(𝑥𝑥)𝑥𝑥 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒lim 𝑥𝑥→0 ln(𝑥𝑥)×𝑥𝑥 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒lim 𝑥𝑥→0 ln 𝑥𝑥 𝑥𝑥−1 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒 lim 𝑥𝑥→0 𝑑𝑑 𝑑𝑑𝑑𝑑ln 𝑥𝑥 𝑑𝑑 𝑑𝑑𝑑𝑑𝑥𝑥−1 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒lim 𝑥𝑥→0 𝑥𝑥−1 −𝑥𝑥−2 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒lim 𝑥𝑥→0 𝑥𝑥 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 𝑒𝑒0 lim 𝑥𝑥→0 𝑥𝑥𝑥𝑥 = 1
7. Why this matters Mathematicians argue this because they need formulas to work. Example The Binomial Theorem … (𝑎𝑎 + 𝑏𝑏)𝑛𝑛 (𝑎𝑎 + 𝑏𝑏)𝑛𝑛= � 𝑘𝑘=0 𝑛𝑛 𝑛𝑛 𝑘𝑘 𝑎𝑎𝑛𝑛−𝑘𝑘𝑏𝑏𝑘𝑘 When a or b = 0, we get 00 in the formula which as to equal 1: (𝑎𝑎 + 0)𝑛𝑛= � 𝑘𝑘=0 𝑛𝑛 𝑛𝑛 𝑘𝑘 𝑎𝑎𝑛𝑛−𝑘𝑘0𝑘𝑘 = 𝑛𝑛 0 𝑎𝑎𝑛𝑛00 + 𝑛𝑛 1 𝑎𝑎𝑛𝑛−101 …
8. Conclusion -Technically, 00 = 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 (like 0 ÷ 0) -In mathematic convention, 00 = 1 -The value for a point arbitrarily close to 𝑥𝑥 = 0 in 𝑦𝑦 = 𝑥𝑥𝑥𝑥 would be 1