Chapter 8: Problem 27
The distributive law \(a(b-c)=a b-a c\) does not hold, in general, if the products are rounded off to a smaller number of digits. To show this in a specific case take \(a=0.22,\) \(b=3.19,\) and \(c=2.17 .\) After each multiplication round off the last digit.
Short Answer
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Question: Prove that the distributive law does not hold in general if the products are rounded off to a smaller number of digits, using the specific numbers \(a=0.22\), \(b=3.19\), and \(c=2.17\).
In this case, we calculated \(a(b-c)\) and \((ab-ac)\) with rounding off, and found that they both approximately equal to \(0.22\). Although they are equal in this specific example, this does not disprove the distributive law in general. Rounding might lead to different results than expected, so it should be taken into consideration when working with rounded values.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Law
The distributive law is a basic property of arithmetic that allows the multiplication of a number by a group of numbers added together to be done individually and then summed up. In mathematical terms, it can be expressed as \( a(b+c) = ab + ac \). This law simplifies the multiplication process and is used extensively in algebra for expanding expressions.
When we apply the distributive law in calculations, we expect the outcomes to be consistent. However, when dealing with numbers that require rounding, especially in the context of limited arithmetic precision, the distributive law may not always hold true. In the given exercise, after rounding, the products \( ab \) and \( ac \) separately and then subtracting \( ac \) from \( ab \) may not yield the same result as multiplying \( a \) by the difference \( b-c \) due to loss of information from rounding. This is an important consideration in numerical analysis and computational mathematics, where tiny errors can accumulate and lead to significant inaccuracies.
To mitigate rounding errors in practical applications, it's crucial to maintain as many significant digits as possible until the final result and to understand that the distributive law should be applied carefully within the context of floating-point arithmetic.
When we apply the distributive law in calculations, we expect the outcomes to be consistent. However, when dealing with numbers that require rounding, especially in the context of limited arithmetic precision, the distributive law may not always hold true. In the given exercise, after rounding, the products \( ab \) and \( ac \) separately and then subtracting \( ac \) from \( ab \) may not yield the same result as multiplying \( a \) by the difference \( b-c \) due to loss of information from rounding. This is an important consideration in numerical analysis and computational mathematics, where tiny errors can accumulate and lead to significant inaccuracies.
To mitigate rounding errors in practical applications, it's crucial to maintain as many significant digits as possible until the final result and to understand that the distributive law should be applied carefully within the context of floating-point arithmetic.
Significant Digits
Significant digits, also known as significant figures, are the digits in a number that carry meaning contributing to its precision. These include all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal part. For example, in the number \( 0.2244 \), all four digits are significant because they provide specific information about the quantity measured.
Understanding significant digits is vital in mathematics and science as they indicate the precision of a measurement or calculation. When rounding numbers in calculations, like the product \( 0.22 \times 1.02 \) in the exercise, which gives \( 0.2244 \), to maintain the original number of significant digits (\(0.22\) has two significant digits), we often round the result to \( 0.22 \). However, rounding can introduce a rounding error, which is the difference between the actual number and the rounded number. This can affect the final outcome of calculations significantly, especially when subsequent mathematical operations are sensitive to small changes in the input values.
In scenarios where high precision is required, keeping track of significant digits is essential for ensuring that the final reported answer reflects the precision of the measurements and the limitations of the arithmetic operations performed.
Understanding significant digits is vital in mathematics and science as they indicate the precision of a measurement or calculation. When rounding numbers in calculations, like the product \( 0.22 \times 1.02 \) in the exercise, which gives \( 0.2244 \), to maintain the original number of significant digits (\(0.22\) has two significant digits), we often round the result to \( 0.22 \). However, rounding can introduce a rounding error, which is the difference between the actual number and the rounded number. This can affect the final outcome of calculations significantly, especially when subsequent mathematical operations are sensitive to small changes in the input values.
In scenarios where high precision is required, keeping track of significant digits is essential for ensuring that the final reported answer reflects the precision of the measurements and the limitations of the arithmetic operations performed.
Arithmetic Precision
Arithmetic precision refers to the level of detail and exactness in carrying out operations and representing numbers in calculations. In mathematical computations, precision dictates how closely a calculation can approach the exact result. The number of significant digits largely determines the precision of a numerical value.
In the context of this exercise, arithmetic precision is particularly important in the multiplication steps. Since we are rounding off to fewer digits, we're artificially limiting the arithmetic precision. This influences the accuracy of the final result. For example, when you multiply \( 0.22 \times 1.02 \) without rounding, you get \( 0.2244 \) – but rounding off reduces the precision to just two significant digits, yielding \( 0.22 \).
Similar rounding in subsequent operations can lead to a compounding of errors, termed cumulative rounding error, which may become significant in long calculations. It's a critical aspect to consider in fields like numerical analysis, finance, and any science or engineering domain where precise calculations are fundamental. To improve the reliability of the results, the general advice is to perform calculations with as much precision as possible and only round off at the final step, ensuring the rounding error is minimized.
In the context of this exercise, arithmetic precision is particularly important in the multiplication steps. Since we are rounding off to fewer digits, we're artificially limiting the arithmetic precision. This influences the accuracy of the final result. For example, when you multiply \( 0.22 \times 1.02 \) without rounding, you get \( 0.2244 \) – but rounding off reduces the precision to just two significant digits, yielding \( 0.22 \).
Similar rounding in subsequent operations can lead to a compounding of errors, termed cumulative rounding error, which may become significant in long calculations. It's a critical aspect to consider in fields like numerical analysis, finance, and any science or engineering domain where precise calculations are fundamental. To improve the reliability of the results, the general advice is to perform calculations with as much precision as possible and only round off at the final step, ensuring the rounding error is minimized.
