Chapter 18, Problem 31E

Explanation of Solution

Representational error:

Representational error is an error which occurs when performing arithmetic operations such that the precision digit produced by the arithmetic calculation is greater than the precision of the machine.

• The addition of $\left(\text{A+B}\right)$ with $\text{C}$ produces the same result as adding A with $\left(\text{B+C}\right)$ according to the associative law.
• But due to the limitation of precision, the result produced by $\text{((A+B)+C)}$ is not same as $\text{(A+(B+C))}$

Example: consider the following equation:

$\text{((A+B)+C)}$        (1)

Substitute the value of $\text{A = –5688 }\times {\text{ 10}}^{\text{3}}$, $B=5689\times 10^3$, and $C=5648\times 10^0$ in the Equation (1)

Initially, compute the sum of the first two values $\left(\text{A+B}\right)$:

  $\begin{array}{l}\text{– 5688 }\times {\text{ 10}}^{\text{3 }}\\ \underset{\_}{\text{ 5689 }\times {\text{ 10}}^{\text{3 }}}\\ \text{ 1 }\times {\text{ 10}}^{\text{3}}\end{array}$        (2)

Here, the value $1\times 10^3$ can be written as $1000\times 10^0$.

Then, compute the addition of the above result of Equation (2) with the value of C, $\text{((A+B)+C)}$

  $\begin{array}{l}\text{1000 }\times {\text{ 10}}^{\text{0}}\\ \underset{\_}{\text{5648 }\times {\text{ 10}}^{\text{0}}}\\ \text{6648 }\times {\text{ 10}}^{\text{0}}\end{array}$        (3)

Consider the following equation:

$\text{(A+(B+C))}$        (4)

Substitute the value of $\text{A }=–5688\times {10}^3$, $B=5689\times 10^3$, and $\text{C = 5648 }\times {\text{ 10}}^{\text{0}}$ in Equation (4).

Compute the sum of the two values $\left(\text{B+C}\right)$:

Here, the value $\text{5689 }\times {\text{ 10}}^{\text{3}}$ can be written as $\text{5689000 }\times {\text{ 10}}^{\text{0}}$.

  $\begin{array}{l}\underset{\rightharpoondown }{\begin{array}{l}5689000\times 10^0\hfill \\ 5648\times 10^0\hfill \end{array}}\\ 5694648\times 10^0{}^{ }\end{array}$        (5)

Here, the value $\text{5694648 }\times {\text{ 10}}^{\text{0}}{}^{}$ can be written as $5694\times 10^3$ when truncating four digits.

Then, compute the addition of the above result of the Equation (5) with the value of A, $(A+(B+C))$

  $\begin{array}{l}5694\times {10}^3\\ \underset{\_}{-5688\times {10}^3}\\ 6\times {10}^3\end{array}$        (6)

Here, the value $\text{6 }\times {\text{ 10}}^{\text{3}}$ can be written as $\text{6000 }\times {\text{ 10}}^{\text{0}}$

Thus, when considering the thousands' place alone the result produced by both Equations (3) and (6) are the same, but the result differs when considering together the thousands, hundreds, tens and one’s place.

Therefore, this type of error is called representational error. It is also called as round-off error.

Cancellation error:

Cancellation error is an error that occurs when performing arithmetic operations between numbers that differ in magnitude.

Example: consider the following equation:

$\left(\text{A + B – C}\right)$        (1)

Substitute the value of $\text{A = 1}$, $\text{B = 0}\text{.00008963}$, and $\text{C = 1}$ in Equation (1).

Here, the value $\text{1}$ can be written as $\text{100000000 }\times {\text{ 10}}^{\text{-8}}$ and the value $\text{0}\text{.00008963}$ can be written as $\text{8963 }\times {\text{ 10}}^{\text{-8}}$

Initially, compute the sum of the first two values $\left(\text{A+B}\right)$:

  $\begin{array}{l}100000000\times {10}^{-8}\\ \underset{\_}{ 8963\times {10}^{-8}}\\ 100008963\times {10}^{-8}\end{array}$        (2)

Here, the value $\text{100008963 }\times {\text{ 10}}^{\text{-8}}$ can be written as $\text{1000 }\times {\text{ 10}}^{\text{-3}}$.

Then, compute the subtraction of the above result of Equation (2) with the value of C, $\text{((A+B)-C)}$.

  $\begin{array}{l}\text{1000 }\times {\text{ 10}}^{\text{-3}}\\ \underset{\_}{\text{1000 }\times {\text{ 10}}^{\text{-3}}}\\ \text{ 0}\end{array}$

In general, the result produced by $\left(\text{A + B – C}\right)$ is $0.00008963$. But, the result produced by the computer is 0, which is not $0.00008963$.

Therefore, this type of error is called cancellation error.

Underflow error:

Underflow error is an error which occurs when the precision digit produced by the arithmetic calculation is too small to represent in the machine.

Example: consider the following equation,

$\text{(A+B)}$        (1)

Substitute the value of $\text{A = 5688}\times {\text{10}}^{\text{-7}}$  and $\text{B = 5689}\times {\text{10}}^{\text{-7}}$ in Equation (1).

Compute the sum of the first two values $\left(\text{A+B}\right)$:

  $\begin{array}{l}5688\times {10}^{-7}\\ \underset{\_}{5689\times {10}^{-7}}\\ 11377\times {10}^{-14}\end{array}$

Here, the value $\text{11377}\times {\text{10}}^{\text{-14}}$ can be written as $\text{1137}\times {\text{10}}^{\text{-13}}$.

The result of (A+B) is $\text{1137}\times {\text{10}}^{\text{-13}}$, which is too small to represent in the machine because the minimum exponent value that can be represented is $\text{-9}$ and the value lesser than the exponent -9 is considered as 0. So the result of A + B is set to 0.

Therefore, this type of error is called underflow error.

Overflow error:

Overflow error is an error which occurs when the precision digit produced by the arithmetic calculation is too large to represent in the machine.

Example: consider the following equation:

$\text{(A+B)}$        (1)

Substitute the value of $\text{A = 5688}\times {\text{10}}^{\text{7}}$  and $\text{B = 5689}\times {\text{10}}^{\text{7}}$ in Equation (1).

Compute the sum of the first two values $\left(\text{A+B}\right)$:

  $\begin{array}{l}5688\times {10}^7\\ \underset{\_}{ 5689\times {10}^7}\\ 11377\times {10}^{14}\end{array}$

Here, the value $\text{11377}\times {\text{10}}^{\text{14}}$ can be written as $\text{1137}\times {\text{10}}^{\text{15}}$.

The result of $\text{(A+B)}$ is $\text{1137}\times {\text{10}}^{\text{15}}$ which is too large to represent in the machine, because the maximum exponent value that can be represented is $\text{+9}$. So an error message can be produced to indicate the error.

Therefore, this type of error is called overflow error.

Explanation of Solution

Interrelated terms:

The above terms, “Representational error”, “Underflow error”, “Overflow error”, “Cancellation error” is interrelated to each other.

• Because, all the terms represent the error which occurs due to the limitation of representing a number in a computer.