Calculating 10^0.1 by Hand

In this video I go over how to manually calculate powers 10^0.1, 10^0.2, 10^0.3, ..., 10^0.9 by hand by first using the approximation 10^0.3 = 2. Calculating such powers by hand directly in the same way for multiplication and long division is not possible and requires the use of logarithms or Taylor series. Nonetheless, we can manually approximate such powers by realizing that 10^0.03 is approximately equal to 2 (with an error of -0.24%). This means we can approximate multiples of 10^0.03 by multiplying by 2. For example 10^0.06 is approximately equal to 4. Note that the error becomes double because we multiplied by 2. To get powers such as 10^0.1, we can multiply 10^(0.03 * 7) = 10^2.1 ≈ 2^7 = 128. Then we can divide both sides by 10^2 = 100 to get an approximation 10^0.1 ≈ 1.28. Applying the correction of 7*0.24% we obtain 10^0.1 ≈ 1.2585 which is very close to the actual number 1.2589.

Video inspiration and calculations obtained from the links below:

- Art of Memory article: https://forum.artofmemory.com/t/calculating-10-0-1-0-2-0-3-etc/29957
- OmegaKlass video request: https://www.bitchute.com/video/9tSG173RqQEC/

Timestamps:

- Intro: 0:00
- Video request by OmegaKlass: 0:38
- Calculating 10^0.01 by hand: 1:06
- 10^0.3 is approximately equal to 2: 2:28
- 10^0.3 = a number multiplied by itself 10 times equals 1000: 3:13
- Error of 0.24%: 6:30
- Approximating multiples of 10^0.3: 7:17
- Divide by powers of 10 to obtain more powers: 8:53
- Summarizing our simple approximations: 16:54
- Note the pattern: Find multiple of 3 that ends with desired power: 18:34
- Applying the correction (note my voice is when I was sick): 22:10
- Table comparison of approximation and actual values: 27:41
- Outro: 30:01

Video notes and playlist:

- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIitoplA_8nRHntU8KSw
- HIVE notes: https://peakd.com/hive-128780/@mes/calculating-10-01-by-hand
- Calculating by Hand playlist: https://www.youtube.com/playlist?list=PL0919AEF28CB3D765 .

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