What are the definitions and examples of approximation error, significant digits, and terminating decimals?

a) The difference between exact and approximated value is called:
The difference between an exact value and an approximated value is known as the approximation error.
The correct option is ii).

$$\boxed{\text{ii) approximation error}}$$

b) Which one has 4 significant digits?
To determine the number of significant digits:

• i) 0.0012: The leading zeros are not significant. The significant digits are 1 and 2. This has 2 significant digits.
• ii) 0.0125: The leading zeros are not significant. The significant digits are 1, 2, and 5. This has 3 significant digits.
• iii) 63.27: All non-zero digits are significant. The significant digits are 6, 3, 2, and 7. This has 4 significant digits.
• iv) 0.356: The leading zero is not significant. The significant digits are 3, 5, and 6. This has 3 significant digits. The correct option is iii).

$$\boxed{\text{iii) } 63.27}$$

c) Which fraction shows the terminating decimal?
A fraction represents a terminating decimal if, when simplified, the prime factors of its denominator are only $2$s and $5$s.

• i) $\frac{6}{5}$: The denominator is 5. Since 5 is a prime factor, this is a terminating decimal ($6 \div 5 = 1.2$).
• ii) $\frac{7}{13}$: The denominator is 13. Since 13 is not 2 or 5, this is a non-terminating, repeating decimal.
• iii) $\frac{5}{3}$: The denominator is 3. Since 3 is not 2 or 5, this is a non-terminating, repeating decimal ($5 \div 3 = 1.666...$).
• iv) $\frac{5}{9}$: The denominator is $9 = 3^2$. Since 3 is not 2 or 5, this is a non-terminating, repeating decimal ($5 \div 9 = 0.555...$). The correct option is i).

$$\boxed{\text{i) } \frac{6}{5}}$$

d) The union of rational and irrational numbers is called:

• Rational numbers are numbers that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
• Irrational numbers are numbers that cannot be expressed as a simple fraction (e.g., $\sqrt{2}$, $\pi$). The set of all rational and irrational numbers together forms the set of real numbers. The correct option is iii).

$$\boxed{\text{iii) real numbers}}$$