Chopping, Rounding Examples
1. Determine the five-digit (a) chopping and (b) rounding values of the irrational number π.
The value of π (pi) is approximately 3.14159265358979323846...
(a) Chopping:
Chopping means truncating or cutting off the decimal part after a certain number of digits. In this case, we want to determine the five-digit chopping value of π.
The first five digits of π are 3.1415. So the five-digit chopping value of π is 3.1415.
(b) Rounding:
Rounding means approximating a number to the nearest value with a specified number of digits. In this case, we want to determine the five-digit rounding value of π.
The sixth digit of π is 9, and since it is greater than or equal to 5, we round up the fifth digit (4) to the next higher value, which becomes 5. Therefore, the five-digit rounding value of π is 3.1416.
2. Suppose that x = 5/7 and y = 1/3 . Use five-digit chopping for calculating x + y, x − y, x × y, and x ÷ y
x + y:
x = 5/7
y = 1/3
x + y = (5/7) + (1/3) = (15/21) + (7/21) = 22/21
Now, using five-digit chopping, we keep only the first five digits after the decimal point:
x + y ≈ 1.04761
x - y:
x = 5/7
y = 1/3
x - y = (5/7) - (1/3) = (15/21) - (7/21) = 8/21
Now, using five-digit chopping, we keep only the first five digits after the decimal point:
x - y ≈ 0.38095
x × y:
x = 5/7
y = 1/3
x × y = (5/7) × (1/3) = 5/21
Now, using five-digit chopping, we keep only the first five digits after the decimal point:
x × y ≈ 0.23809
x ÷ y:
x = 5/7
y = 1/3
x ÷ y = (5/7) ÷ (1/3) = (5/7) × (3/1) = 15/7
Now, using five-digit chopping, we keep only the first five digits after the decimal point:
x ÷ y ≈ 2.14285
Please note that using five-digit chopping means rounding off the result to five decimal places after each calculation.
3. Let p = 0.54617 and q = 0.54601. Use four-digit arithmetic to approximate p − q and determine the absolute and relative errors using (a) rounding and (b) chopping.
Rounding
Round p and q to four digits.
p = 0.5462
q = 0.5460
Subtract the rounded values of p and q.
p - q = 0.5462 - 0.5460 = 0.0002
Calculate the absolute error.
Absolute error = |0.0002 - 0.00016| = 0.00004
Calculate the relative error.
Relative error = 0.00004 / 0.00016 = 0.25
Chopping
Chop p and q to four digits.
p = 0.5461
q = 0.5460
Subtract the chopped values of p and q.
p - q = 0.5461 - 0.5460 = 0.0001
Calculate the absolute error.
Absolute error = |0.0001 - 0.00016| = 0.00006
Calculate the relative error.
Relative error = 0.00006 / 0.00016 = 0.375
As you can see, the absolute and relative errors are slightly larger when rounding than when chopping. This is because rounding rounds off the values of p and q to the nearest four digits, while chopping truncates the values of p and q to the nearest four digits. Therefore, chopping is a more accurate method of rounding numbers than rounding.
4. Perform the following computations (i) exactly, (ii) using three-digit chopping arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Compute the relative errors in parts (ii) and (iii). d. (1/3 +3/11)−(3/20)
(i) Exact computation:
To find (1/3 + 3/11) - (3/20), we can simply calculate the expression:
(1/3 + 3/11) - (3/20) = 0.333333... + 0.272727... - 0.15
= 0.606060... - 0.15
= 0.456060...
(ii) Three-digit chopping arithmetic:
Using three-digit chopping arithmetic, we truncate each fraction to three digits before performing the operations.
1/3 = 0.333
3/11 = 0.273
3/20 = 0.15
Now, we perform the subtraction:
(0.333 + 0.273) - 0.15 = 0.606 - 0.15 = 0.456
(iii) Three-digit rounding arithmetic:
Using three-digit rounding arithmetic, we round each fraction to three digits before performing the operations.
1/3 = 0.333
3/11 = 0.273
3/20 = 0.15
Now, we perform the subtraction:
(0.333 + 0.273) - 0.15 = 0.606 - 0.15 = 0.456
(iv) Compute the relative errors in parts (ii) and (iii):
The relative error for chopping arithmetic:
Relative error = |Exact value - Chopped value| / |Exact value|
Relative error = |0.456060... - 0.456| / |0.456060...|
Relative error ≈ 0
The relative error for rounding arithmetic:
Relative error = |Exact value - Rounded value| / |Exact value|
Relative error = |0.456060... - 0.456| / |0.456060...|
Relative error ≈ 0
In this case, the relative errors for both the chopping and rounding arithmetic are approximately 0. This means that the approximate results obtained using three-digit chopping and rounding are very close to the exact result.