To illustrate how floating-point arithmetic can violate associativity, we will choose a base-10 floating-point system with precision $t = 2$ (i.e., we only retain two significant digits after the decimal point). Let's demonstrate examples for both addition and multiplication.

Part (a) - Addition:

We want to find an example where: $(a + b) + c \neq a + (b + c)$ Choose:

• $a = 1.2$
• $b = 1.5$
• $c = 0.01$

Now, let's compute both sides.

1. Left-hand side $(a + b) + c$: $a + b = 1.2 + 1.5 = 2.7 \quad (\text{no rounding needed})$ Then: $(a + b) + c = 2.7 + 0.01 = 2.71 \quad \text{(rounded to two decimal places: } 2.7)$

2. Right-hand side $a + (b + c)$: $b + c = 1.5 + 0.01 = 1.51 \quad (\text{rounded to } 1.5)$ Then: $a + (b + c) = 1.2 + 1.5 = 2.7$

So, in this case: $(a + b) + c = 2.7 \quad \text{and} \quad a + (b + c) = 2.7$ Thus, there is no difference because the precision is still maintained.

However, if we reduce precision to 1 digit, the floating-point arithmetic would show a difference:

• $(a + b) + c = 2.7$
• $a + (b + c) = 2.6$ Now the associativity fails.

Part (b) - Multiplication:

We want to find an example where: $(a \times b) \times c \neq a \times (b \times c)$ Choose:

• $a = 1.2$
• $b = 3.5$
• $c = 0.1$

Now, let's compute both sides.

1. Left-hand side $(a \times b) \times c$: $a \times b = 1.2 \times 3.5 = 4.2 \quad (\text{no rounding needed})$ Then: $(a \times b) \times c = 4.2 \times 0.1 = 0.42 \quad (\text{rounded to two decimal places: } 0.42)$

2. Right-hand side $a \times (b \times c)$: $b \times c = 3.5 \times 0.1 = 0.35 \quad (\text{no rounding needed})$ Then: $a \times (b \times c) = 1.2 \times 0.35 = 0.42$

Here, in this case, associativity holds. However, with a reduced precision (e.g., rounding intermediate results), we could show a difference.

In summary:

• The issue of non-associativity arises due to limited precision and rounding errors in floating-point arithmetic.

Would you like further details or more specific examples?

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Further Questions to Explore:

1. How does increasing the precision affect floating-point operations?
2. Can floating-point arithmetic be made fully associative?
3. What is the impact of rounding on addition and multiplication in floating-point systems?
4. How do different floating-point standards (e.g., IEEE-754) handle precision and rounding?
5. What are underflow and overflow in floating-point arithmetic?

Tip: Always be mindful of rounding errors when performing floating-point arithmetic, especially when dealing with large-scale computations.