Q.Write down the characteristic of the common logarithm of each of the following numbers:

The characteristic depends only on the position of the decimal point, found by the digit-count rule for numbers $\geq 1$ and the leading-zeros rule for numbers $<1$.

1. 4567 — a number $\geq 1$ with 4 digits before the decimal point. Characteristic $= 4 - 1 = 3$.
2. 45.67 — a number $\geq 1$ with 2 digits before the decimal point. Characteristic $= 2 - 1 = 1$.
3. 0.4567 — a number $< 1$ with no zeros immediately after the decimal point before the first significant digit (4). Characteristic $= -[0 + 1] = -1$, written $\bar{1}$.
4. 0.004567 — a number $< 1$ with two zeros immediately after the decimal point before the first significant digit. Characteristic $= -[2 + 1] = -3$, written $\bar{3}$. …

Cross-check against the powers of 10 the numbers lie between. (i) $4567$ lies between $10^{3}=1000$ and $10^{4}=10000$, so its characteristic is 3 ✓. (ii) $45.67$ lies between $10^{1}$ and $10^{2}$, characteristic 1 ✓. (iii) $0.4567$ lies between $10^{-1}=0.1$ and $10^{0}=1$, cha …

For numbers less than 1, students often forget the "+1" and give the characteristic of 0.004567 as $\bar{2}$ (just the two zeros) instead of $\bar{3}$. The rule counts the leading zeros plus one. Another slip is writing a negative characteristic as an o …