TLDR: don’t bother memorizing half powers. Instead, memorize a few logarithms.
I’m an engineer who does a lot of arithmetic in my head, because I lie awake at night designing stuff. I agree being able to do half-power-of-ten math in your head is very useful. But I didn’t have to drill it, because I can do it by converting to logs, adding, and converting back (as Rohin Shah suggests). And I’ve done that enough that I’ve memorized all the common combinations without conscious effort.
A few years ago I used a set of flash cards and spaced repetition drills to memorize the log_10 of a a half-dozen common numbers. That’s been moderately useful for multiplication, powers and roots. I’m not sure if it has paid off in net. I can do arithmetic pretty good by other methods, and the experience of trying to cudgel numbers into my brain was not fun for me. But some people love their Anki cards, and for them, logs are probably a good thing to learn.
Useful facts: sqrt(10) = 3.16. Logs of 2, 4, 8 are .30, .60, 90. Log_10(5)=.70. If you need more than two digits, use a computer.
IANAE but what I use is log_10 by increments of .1. The starting point is that 10^.1 is roughly 5⁄4. (It’s actually ~1.259 rather than 1.25.) You get, basically,
1
1.25
1.57
2
2.5
3.1
4
5
6.3
8
10
and you can pretty closely regenerate this with 10^.1 = 5⁄4, accelerated by remembering 10^.3 is 2 (which pins down 4 and 8).
TLDR; you probably already know that 2^10=1024, use this to derive powers of 10 instead of memorizing!
https://en.wikipedia.org/wiki/Renard_series, which were designed in the 1870s to be convenient for the officers and engineers of the French Army without a slide rule or a log table, are based on 5th and 10th roots of 10. 1024 being quite close to 1000 means that 3√2 is very close to 100.1, and this allows you to quickly derive R10 numbers without pen and paper.
I have used the algorithm for so long that it has become almost unconscious so I had Gemini write it out:
Mental Algorithm for R10 Numbers (sorry for poor formatting, it doesn’t copypaste neatly, I only fixed manually where it doesn’t read well)
In the R10 series, every step increases the value by a factor of $\approx 1.26$. However, since 210≈103: 103/10≈2 This gives you the Golden Rule of R10: * Add 3 to the Index → Multiply Value by 2 * Subtract 3 from the Index → Divide Value by 2
### The Algorithm: The Three Strands To find any R10 number mentally, you don’t calculate them sequentially. Instead, you split the numbers 0–10 into three “strands” based on the anchors you already know: **1**, **8**, and **10**.
#### Strand A: The Powers of 2 (Indices 0, 3, 6, 9) Start at **1** and double it every 3 steps. * R10(**0**) = **1.0** * R10(**3**) = **2.0** * R10(**6**) = **4.0** * R10(**9**) = **8.0**
#### Strand B: The Halving from 10 (Indices 10, 7, 4, 1) Start at **10** and halve it every 3 steps (going backwards). * R10(**10**) = **10.0** * R10(**7**) = **5.0** * R10(**4**) = **2.5** * R10(**1**) = **1.25**
#### Strand C: The “80% Rule” (Indices 8, 5, 2) This is the hardest strand because it doesn’t land on a clean integer. We derive this by starting at R10(9), which we know is **8.0**, and going **down 1 step**. Mathematically, going down 1 step is dividing by $1.2589...$, which is almost exactly multiplying by **0.8**. * Start at R10(9) = 8.0. * **R10(8)** $\approx 8.0 \times 0.8 =$ **6.4** (Anchor) * Now, apply the “Subtract 3 is Half” rule: * **R10(5)** $\approx 6.4 / 2 =$ **3.2** * **R10(2)** $\approx 3.2 / 2 =$ **1.6**
### Summary Table (Mental vs Actual)
By using this mental model (Doubling, Halving, and the 0.8 factor), your approximations are incredibly close to the standard values.
### Quick Reference for Your Brain 1. **0, 3, 6, 9:** Just say **1, 2, 4, 8**. 2. **1, 4, 7:** Start at 10 and halve backwards ($10 \to 5 \to 2.5 \to 1.25$). 3. **2, 5, 8:** Remember **6.4** (from $8 \times 8$), then halve backwards (6.4→3.2→1.6).
For the 5th R10 number you can also use the coincidence that square root of 10 is close to pi (used in antiquity to approximate pi), or for the 8th number you can use 2.52=(2+0.5)∗(3−0.5)=2∗3+1.5−1−0.25=6.25 (I personally have just memorized it in middle school), but neither is really necessary for mental calculations
TLDR: don’t bother memorizing half powers. Instead, memorize a few logarithms.
I’m an engineer who does a lot of arithmetic in my head, because I lie awake at night designing stuff. I agree being able to do half-power-of-ten math in your head is very useful. But I didn’t have to drill it, because I can do it by converting to logs, adding, and converting back (as Rohin Shah suggests). And I’ve done that enough that I’ve memorized all the common combinations without conscious effort.
A few years ago I used a set of flash cards and spaced repetition drills to memorize the log_10 of a a half-dozen common numbers. That’s been moderately useful for multiplication, powers and roots. I’m not sure if it has paid off in net. I can do arithmetic pretty good by other methods, and the experience of trying to cudgel numbers into my brain was not fun for me. But some people love their Anki cards, and for them, logs are probably a good thing to learn.
Useful facts: sqrt(10) = 3.16. Logs of 2, 4, 8 are .30, .60, 90. Log_10(5)=.70. If you need more than two digits, use a computer.
IANAE but what I use is log_10 by increments of .1. The starting point is that 10^.1 is roughly 5⁄4. (It’s actually ~1.259 rather than 1.25.) You get, basically,
1 1.25 1.57 2 2.5 3.1 4 5 6.3 8 10
and you can pretty closely regenerate this with 10^.1 = 5⁄4, accelerated by remembering 10^.3 is 2 (which pins down 4 and 8).
TLDR; you probably already know that 2^10=1024, use this to derive powers of 10 instead of memorizing!
https://en.wikipedia.org/wiki/Renard_series, which were designed in the 1870s to be convenient for the officers and engineers of the French Army without a slide rule or a log table, are based on 5th and 10th roots of 10. 1024 being quite close to 1000 means that 3√2 is very close to 100.1, and this allows you to quickly derive R10 numbers without pen and paper.
I have used the algorithm for so long that it has become almost unconscious so I had Gemini write it out:
Mental Algorithm for R10 Numbers (sorry for poor formatting, it doesn’t copypaste neatly, I only fixed manually where it doesn’t read well)
In the R10 series, every step increases the value by a factor of $\approx 1.26$.
However, since 210≈103:
103/10≈2
This gives you the Golden Rule of R10:
* Add 3 to the Index → Multiply Value by 2
* Subtract 3 from the Index → Divide Value by 2
### The Algorithm: The Three Strands
To find any R10 number mentally, you don’t calculate them sequentially. Instead, you split the numbers 0–10 into three “strands” based on the anchors you already know: **1**, **8**, and **10**.
#### Strand A: The Powers of 2 (Indices 0, 3, 6, 9)
Start at **1** and double it every 3 steps.
* R10(**0**) = **1.0**
* R10(**3**) = **2.0**
* R10(**6**) = **4.0**
* R10(**9**) = **8.0**
#### Strand B: The Halving from 10 (Indices 10, 7, 4, 1)
Start at **10** and halve it every 3 steps (going backwards).
* R10(**10**) = **10.0**
* R10(**7**) = **5.0**
* R10(**4**) = **2.5**
* R10(**1**) = **1.25**
#### Strand C: The “80% Rule” (Indices 8, 5, 2)
This is the hardest strand because it doesn’t land on a clean integer.
We derive this by starting at R10(9), which we know is **8.0**, and going **down 1 step**.
Mathematically, going down 1 step is dividing by $1.2589...$, which is almost exactly multiplying by **0.8**.
* Start at R10(9) = 8.0.
* **R10(8)** $\approx 8.0 \times 0.8 =$ **6.4** (Anchor)
* Now, apply the “Subtract 3 is Half” rule:
* **R10(5)** $\approx 6.4 / 2 =$ **3.2**
* **R10(2)** $\approx 3.2 / 2 =$ **1.6**
### Summary Table (Mental vs Actual)
By using this mental model (Doubling, Halving, and the 0.8 factor), your approximations are incredibly close to the standard values.
| Index | Mental Derivation | Approx Value | Actual R10 Value |
| :--- | :--- | :--- | :--- |
| **0** | Base | **1.00** | 1.00 |
| **1** | $10 \div 8$ | **1.25** | 1.25 |
| **2** | $3.2 \div 2$ | **1.60** | 1.60 |
| **3** | $1 \times 2$ | **2.00** | 2.00 |
| **4** | $5 \div 2$ | **2.50** | 2.50 |
| **5** | $6.4 \div 2$ | **3.20** | 3.15 |
| **6** | $2 \times 2$ | **4.00** | 4.00 |
| **7** | $10 \div 2$ | **5.00** | 5.00 |
| **8** | $8 \times 0.8$ | **6.40** | 6.30 |
| **9** | $4 \times 2$ | **8.00** | 8.00 |
| **10** | Base | **10.00** | 10.00 |
### Quick Reference for Your Brain
1. **0, 3, 6, 9:** Just say **1, 2, 4, 8**.
2. **1, 4, 7:** Start at 10 and halve backwards ($10 \to 5 \to 2.5 \to 1.25$).
3. **2, 5, 8:** Remember **6.4** (from $8 \times 8$), then halve backwards (6.4→3.2→1.6).
For the 5th R10 number you can also use the coincidence that square root of 10 is close to pi (used in antiquity to approximate pi), or for the 8th number you can use 2.52=(2+0.5)∗(3−0.5)=2∗3+1.5−1−0.25=6.25 (I personally have just memorized it in middle school), but neither is really necessary for mental calculations