Section 22–2 The inverse operations

(Subtraction & division / Root & logarithm / Class of objects)

In this section, Feynman explains inverse operations from the perspectives of “subtraction & division,” “root & logarithm,” and the class of objects.

1. Subtraction & division:

“If a + b = c, b is defined as c − a, which is called subtraction. The operation called division is also clear: if ab = c, then b = c/a defines division (Feynman et al., 1963, section 22–2 The inverse operations).”

According to Feynman, the direct operations are addition, multiplication, and raising to a power. He defines two inverse operations as follows: 1. Subtraction: if a + b = c, then b is defined as c – a. 2. Division: if a ´ b = c, then b is defined as c/a. Alternatively, one may define a subtraction as an addition of an inverse, a − b = a + (−b). More important, we can define an inverse operation as an operation that undoes what was done by the previous operation. Thus, subtraction is an inverse operation of addition, and addition is an inverse operation of subtraction. Similarly, division is an inverse operation of multiplication, and multiplication is an inverse operation of division. To be precise, we should also define a, b, and c in terms of a class of objects.

To be comprehensive, one may discuss properties of subtraction and division with regard to commutativity and associativity. Specifically, subtraction does not have commutative property because 1 – 0 = 1 does not equal to 0 – 1 = –1; i.e., the order of numbers affects the outcome. Subtraction also does not have associative property because (1 – 1) – 1 = –1 does not equal to 1 – (1 – 1) = +1; i.e., the order of operations affects the outcome. Furthermore, division does not have commutative property because 1 ¸ 2 = 0.5 does not equal to 2 ¸ 1 = 2. Division also does not have associative property because (1 ¸ 2) ¸ 2 = 0.25 does not equal to 1 ¸ (2 ¸ 2) = 1. In summary, subtraction and division do not have commutative property and associative property.

2. Root & logarithm:

“Because b^a and a^b are not equal, there are two inverse problems associated with powers… (Feynman et al., 1963, section 22–2 The inverse operations).”

Feynman first poses an inverse problem: if b^a = c, what is b? In this case, b = _a√c and it is called the ath root of c. This is similar to the question: “What integer, raised to the third power, equals 8?” The other inverse problem is: if a^b = c, what is b? This case has a different answer: b = log_a c and it is similar to the question: “To what power must we raise 2 to get 8?” To distinguish the two problems, we may use the terms exponential function and power function. That is, the exponential function with base a (e.g., f(x) = a^x) and the logarithm function (e.g., g(x) = log_a x) with base a are inverse functions of each other. One may let students show that f[g(x)] = a^{log x} = g[f(x)] = log a^x = x. Similarly, the power function (e.g., u(x) = xⁿ) and the root function (e.g., v(x) = _n√x = x^1/n) are inverse functions of each other: u[v(x)] = (x^1/n)ⁿ = v[u(x)] = (xⁿ)^1/n = x.

Perhaps Feynman should have discussed more properties of power and logarithm operation. To illustrate how the power operation does not have commutative property and associative property, we can use the symbol ^ such that a ^ b = a^b. First, the power operation does not have commutative property because 1 ^ 2 = 1² = 1 does not equal to 2 ^ 1 = 2¹ = 2. Second, the power operation does not have associative property because (2 ^ 1) ^ 2 = 2² = 4 does not equal to 2 ^ (1 ^ 2) = 2¹ = 2. By using the similar method, we can prove that the root operation (Öx) and logarithm operation (log x) are not commutative and not associative. One may let students prove the following important properties: 1. Product property: log_b xy = log_b x + log_b y. 2. Quotient property: log_b x/y = log_b x – log_b y. 3. Power property: log_b xⁿ = n log_b x.

3. Class of objects:

“We are going to discuss whether or not we can broaden the class of objects which a, b, and c represent so that they will obey these same rules… (Feynman et al., 1963, section 22–2 The inverse operations).”

Feynman says that the relationships or rules are correct for integers since they follow from the definitions of addition, multiplication, and raising to a power. In division, the rules are not completely correct because mathematicians would emphasize that a division of a number by zero is undefined. On the other hand, the logarithm of zero is also undefined. Curiously, Feynman continues saying that we can broaden the class of objects…, although the processes for a + b, and so on, will not be definable. It is likely that he was not specifically referring to a + b, but log_a b and _aÖ b that cannot be defined without using irrational numbers or complex numbers.

In What Do You Care What Other People Think?, Feynman (1988) writes: “I learned algebra, fortunately, not by going to school, but by finding my aunt’s old schoolbook in the attic, and understanding that the whole idea was to find out what x is - it doesn't make any difference how you do it. For me, there was no such thing as doing it ‘by arithmetic,’ or doing it ‘by algebra.’ ‘Doing it by algebra’ was a set of rules which, if you followed them blindly, could produce the answer: ‘subtract 7 from both sides; if you have a multiplier, divide both sides by the multiplier,’ and so on - a series of steps by which you could get the answer if you didn’t understand what you were trying to do. The rules had been invented so that the children who have to study algebra can all pass it. And that’s why my cousin was never able to do algebra (Feynman, 1988, p. 17).” It is possible that Feynman’s understanding of algebra may not be conventional.

Questions for discussion:

1. How would you define inverse operations?

2. How would you list the properties of subtraction, division, root, and logarithm?

3. Is the class of objects for addition the same as subtraction?

The moral of the lesson: subtraction, division, root, and logarithm are the inverse operations for addition, multiplication, power, and exponent respectively; in a sense, direct operations and inverse operations are opposite operations that undo each other.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Feynman, R. P. (1988). What Do You Care What Other People Think?. New York: W W Norton.