e) The recurring decimal is:

e) The recurring decimal is:
A recurring decimal is a decimal in which one or more digits repeat infinitely.

• i) 3.4444...: The digit 4 repeats indefinitely. This is a recurring decimal.
• ii) 2.34178...: The ellipsis indicates continuation, but no repeating pattern is shown.
• iii) 0.23142...: No repeating pattern is shown.
• iv) 3.0014...: No repeating pattern is shown. The correct option is i).

$$\boxed{\text{i) } 3.4444...}$$

f) Multiplicative identity for the set of real numbers is:
The multiplicative identity is the number that, when multiplied by any real number, leaves the number unchanged. For real numbers, this is 1.

• i) 0: This is the additive identity.
• ii) 1: For any real number $x$, $x \times 1 = x$.
• iii) $-1$: Multiplying by $-1$ changes the sign of the number.
• iv) $\frac{1}{2}$: This is not the multiplicative identity. The correct option is ii).

$$\boxed{\text{ii) } 1}$$

g) The additive inverse of $\frac{3}{8}$ is:
The additive inverse of a number $x$ is the number $-x$ such that $x + (-x) = 0$.
For $\frac{3}{8}$, the additive inverse is $-\frac{3}{8}$.

• i) $-\frac{8}{3}$: This is the negative reciprocal.
• ii) $-\frac{3}{8}$: This is the additive inverse.
• iii) $\frac{3}{8}$: This is the original number.
• iv) 0: This is the additive identity. The correct option is ii).

$$\boxed{\text{ii) } -\frac{3}{8}}$$

h) Which of these is an irrational number?
An irrational number is a number that cannot be expressed as a simple fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Square roots of non-perfect squares are irrational.

• i) $\sqrt{5}$: Since 5 is not a perfect square, $\sqrt{5}$ is an irrational number.
• ii) $\sqrt{9} = 3$: This is a rational number ($3 = \frac{3}{1}$).
• iii) $\sqrt{4} = 2$: This is a rational number ($2 = \frac{2}{1}$).
• iv) $\sqrt{36} = 6$: This is a rational number ($6 = \frac{6}{1}$). The correct option is i).

$$\boxed{\text{i) } \sqrt{5}}$$