Slide rules, logarithms, and analog computers

academics math

Growing up, one of my favorite films was Studio Ghibli’s The Wind Rises—an animated historical drama about a 20th-century Japanese engineer named Jiro Horikoshi. Each time I rewatched it, I was always intrigued by a device that Jiro used for performing calculations. It consisted of two wooden rulers, with the top one able to slide freely. Somehow, in what seemed like magic, Jiro could slide the middle ruler and multiply any two numbers together, say 2.18 x 3.62, and by looking at the image below, obtain the correct answer of 7.89.

Except that it wasn’t magic; it was math.

The simple, clever math that makes a slide rule capable of multiplication is in fact the basis of an analog computer.

Exponentiation and Logarithms

Consider a number denoted by the letter b. When you multiply b by itself, n times, you are performing an operation called exponentiation. Specifically, we can write b × b × … × bn times = bn

We call b the ‘base’ and n the ‘exponent.’  For example, let the base be 10 and the exponent be 3. Since 1000 = 10 × 10 × 10 = 103, we say that 10 ‘raised to the power of 3’ is 1000. 

A number that is multiplied by itself has the interesting property that

bn x bm = b × b ×… × bn times × b × b ×… × bm times = b × b × … × b n + m times = b n + m  

An example of this is 101 x 102 = 10 × 100 = 1000 = 103. Notice, this property allows us to relate the multiplication of two terms with the addition of exponents. Hold on to this fact; it will be crucial shortly.

We are now ready to define a logarithm, which is the inverse function to exponentiation. A logarithm answers the question: ‘how many times must you multiply b by itself to obtain the number x?’  Equivalently, if we choose x and want to know the n for which  x = bn,  then the answer is  n=(x) . If you plug in our solution for n in the expression for x, you should see that x = bx; the logarithm undoes the exponentiation.

Now the fun part: for our property  bn x bm=bn+m, let n=(x) and m=(y) . With this replacement, we find that

b(x × y) = x × y = bx x by =b(x) +(y)

The base for the left and right sides is the same, so for them to be equal the exponents must be equal. We arrive at the expression that allows a slide rule to function:

(x × y) =(x) +(y) .

Notice, we have related multiplication again with addition.

Slide rule mathematics

Let’s examine the expression above carefully. If we could 1) calculate the logarithms of x and y, 2) add them, and 3) somehow undo the logarithm of the sum, we would find the value of  x × y! A slide rule performs these three steps.

For the first step, notice that the digits of a slide rule are different from a typical ruler:

For instance, the separation between 1 and 2 is larger than the separation between 2 and 3. This is because the digits are arranged along a logarithmic scale. The full length of the slide ruler is logb10 - logb1 = logb(10/1) = logb10 , which for most slide rules is 10 inches. Exponentiating both sides yield 10 = b10, such that  b = 1.26. As a result, the distance between 1 and 2 is 3.01 inches, while the distance between 2 and 3 is 1.76 inches.

The power of the logarithmic scale is that (x) and (y) are calculated for us. 

If we pick a value x on the top ruler, say 3.62, and a value y on the bottom ruler, say 2.18,

then their logarithm is simply the physical distance from the left side of the ruler.

For the second step, we simply push the top ruler by the distance (y) by aligning the left side of the top ruler with the position y of the bottom ruler: 

As is illustrated above, this push allows us to create the physical distance corresponding to (x) +(y) between the left side of the bottom ruler and the position of x on the top ruler.

Finally, we look at the value on the bottom ruler beneath x. This value is the product of x and y, since by converting between physical distance to the ruler, we undo the logarithm (x × y) and, at last, obtain x × y.

Slide rule considerations

To strengthen your understanding, I provide a second use of the slide rule to compute 1.5 x 5 = 7.5:

Sometimes, though, the slide rule is too short to perform a calculation. Consider the multiplication of 4 and 5 to produce 20:

Is this calculation not possible with this instrument?

Fortunately, it is possible. Instead of the image above, we can position the right side of the top ruler on y = 4:

The answer is 2, which as usual is given below the x = 5. Try to think about this for a second. Why would repositioning the top ruler lead to an answer that is 10 times smaller than we expect? The answer is that we have shifted the top ruler by its full length, 10 . As a result, we subtracted 10 from 20 , giving logb20 - logb10 = logb(20/10) = 2 . This shows that it is possible to multiply any two numbers using a slide rule, if we keep track of missing factors of 10. 

Analog computers

To recap, the slide rule simplifies calculations by replacing multiplication with addition. What I find incredible is that the slide rule is only one of many objects that use physical phenomena to simply math. These objects are called analog computers, of which the slide rule is one of the simplest examples. 

I’ll leave you with an additional example of an analog computer. In my research I study atoms with lasers, and I frequently send light through lenses. It can be shown that lenses are analog computers capable of modifying the light that passes through them to perform a mathematical operation called a Fourier transform. But showing this, and understanding what a Fourier transform is, is a topic for another day.

Comments

topicTopics
academics study skills MCAT medical school admissions SAT expository writing college admissions English MD/PhD admissions GMAT LSAT GRE writing strategy chemistry physics math biology ACT graduate admissions language learning law school admissions test anxiety interview prep MBA admissions academic advice premed homework help personal statements AP exams career advice creative writing MD study schedules summer activities Common Application history test prep philosophy computer science secondary applications organic chemistry economics supplements PSAT admissions coaching grammar law statistics & probability psychology ESL research 1L CARS SSAT covid-19 legal studies logic games reading comprehension dental admissions mathematics USMLE Spanish calculus engineering parents Latin verbal reasoning DAT case coaching excel mentorship political science French Linguistics Tutoring Approaches academic integrity chinese AMCAS DO MBA coursework PhD admissions Social Advocacy admissions advice biochemistry classics diversity statement genetics geometry kinematics medical school mental health quantitative reasoning skills time management Anki English literature IB exams ISEE MD/PhD programs algebra algorithms art history artificial intelligence astrophysics athletics business business skills careers cold emails data science internships letters of recommendation poetry presentations resume science social sciences software engineering study abroad tech industry trigonometry work and activities 2L 3L Academic Interest DMD EMT FlexMed Fourier Series Greek Health Professional Shortage Area Italian Lagrange multipliers London MD vs PhD MMI Montessori National Health Service Corps Pythagorean Theorem Python STEM Sentence Correction Step 2 TMDSAS Zoom acids and bases amino acids analysis essay architecture argumentative writing brain teaser campus visits cantonese capacitors capital markets cell biology central limit theorem chemical engineering chess chromatography class participation climate change clinical experience community service constitutional law consulting cover letters curriculum demonstrated interest dental school distance learning electricity and magnetism enrichment european history executive function finance first generation student freewriting fun facts functions gap year genomics harmonics health policy history of medicine history of science hybrid vehicles hydrophobic effect ideal gas law induction information sessions institutional actions integrated reasoning intern international students investing investment banking lab reports logic mandarin chinese mba mechanical engineering medical physics meiosis microeconomics mitosis music music theory neurology neuroscience office hours operating systems organization pedagogy phrase structure rules plagiarism pre-dental proofs pseudocode psych/soc quantum mechanics resistors resonance revising scholarships school selection simple linear regression slide decks sociology software stem cells stereochemistry study spots synthesis teaching technical interviews transfer typology units virtual interviews writer's block writing circles