ALGEBRA OF LOGIC
We turn now to the algebra of logic. The first half of [Chapter 9], “Reasoning” (through the section “Logical-Truth Calculation by Algebra”), introduces this subject. There the terms truth values, truth tables, logical connectives, and algebra of logic are explained. The part of [Chapter 3], “A Machine That Will Think,” that discusses the operations greater-than and selection, also explains some of the algebra of logic. It introduces, for example, the formula
p = T(a > b) = 1, 0
This is a way of saying briefly that the truth value of the statement “a is greater than b” equals p; p is 1 if the statement is true and 0 if the statement is false. The truth value 1 corresponds with “yes.” The truth value 0 corresponds with “no.”
With mechanical brains we are especially interested in handling mathematics and logic without any sharp dividing line between them. For example, suppose that we have a register in which a ten-digit number like 1,765,438,890 may be stored. We should be able to use that register to store a number consisting of only 1’s and 0’s, like 1,100,100,010. Such a number may designate the answers to 10 successive questions: yes, yes, no, no, yes, no, no, no, yes, no. Or it may tell 10 successive binary digits. The register then is three times as useful: it can store either decimal numbers or truth values or binary digits. We need, of course, a way to obtain from the register any desired digit. For this purpose we may have two instructions to the machine: (1) read the left-hand end digit; (2) shift the number around in a circle. The second instruction is the same as multiplying by 10 and then putting the left-most digit at the right-hand end. For example, suppose that we want the 3rd digit from the left in 1,100,100,010. The result of the first circular shift is 1,001,000,101; the result of the second circular shift is 0,010,001,011; and reading the left-most digit gives 0. A process like this has been called extraction and is being built into the newest mechanical brains.
Using truth values, we can put down very neatly some truths of ordinary algebra. For example:
- (the absolute value of a) =
- a × (the truth of a greater than or equal to 0)
- - a × (the truth of a less than 0)
|a| = a · T(a ≥ 0) - a · T(a < 0)
For another example:
- Either a is greater than b,
- or else a equals b,
- or else a is less than b
T(a > b) + T(a = b) + T(a < b) = 1
Many common logical operations, like selecting and comparing, and the behavior of many simple mechanisms, like a light or a lock, can be expressed by truth values. [Chapter 4], on punch-card mechanisms, contains a number of examples.
pronoun, variable
In ordinary language, a pronoun, like “he,” “she,” “it,” “the former,” “the latter,” is a word that usually stands for a noun previously referred to. A pronoun usually stands for the last preceding noun that the grammar allows. In mathematics, a variable, like “a,” “b,” “x,” “m₁,” “m₂” closely resembles a pronoun in ordinary language. A variable is a symbol that usually stands for a number previously referred to, and usually it stands for the same number throughout a particular discussion.
multiplicand, dividend, augend, etc.
| In the Equation | The Name of a is: | The Name of b is: | The Name of c is: |
|---|---|---|---|
| a + b = c | augend | addend | sum |
| a - b = c | minuend | subtrahend | remainder |
| a × b = c | multiplicand | multiplier | product |
| a ÷ b = c | dividend | divisor | quotient |
Augend and addend are names of registers in the Harvard Mark II calculator ([see Chapter 10]).
subtraction by adding, nines complement
Two digits that add to 9 (0 and 9, 1 and 8, 2 and 7, 3 and 6, 4 and 5) are called nines complements of each other. The nines complement of a number a is the number b in which each digit of b is the nines complement of the corresponding digit of a; for example, the nines complement of 173 is 826. Ordinary subtraction is the same as addition as of the nines complement, with a simple correction; for example, 562 less 173 (equal to 389) is the same as 562 plus 826 (equal to 1388) less 1000 plus 1.
end-around-carry
The correction “less 1000 plus 1” of the foregoing example may be thought of as carrying the 1 (in the result 1388) around from the left-hand end to the right-hand end, where it is there added. So the 1 is called end-around-carry.
tens complement
Two digits that add to 10 are called tens complements of each other. The tens complement of a number a, however, is equal to the nines complement of the number plus 1. For example, the tens complement of 173 is 827. When subtracting by adding a tens complement, the left-most digit 1 in the result is dropped. For example, 562 less 173 (equal to 389) is the same as 562 plus 827 (equal to 1389) less 1000.
power, square, cube, reciprocal, etc.
A power of any number a is a multiplied by itself some number of times. a × a × a ... × a where a appears b times is written aᵇ and is read a to the bth power. a², a to the 2nd power, is a × a and is called a squared or the square of a. a³, a to the 3rd power, a × a × a, is called a cubed, or the cube of a. a⁰, a to the zero power, is equal to 1 for every a. a¹, a to the power 1, is a itself. The first power is often called linear. a to some negative power is the same as 1 divided by that power; that is, a⁻ᵇ = 1/aᵇ. a⁻¹, a to the power minus 1, is 1/a, and is called the reciprocal of a. a¹ᐟ², a to the one-half power, is a number c such that c × c = a, and is called the square root of a and often denoted by √a.
table, tabular value, argument, etc.
An example of a table is:
| 0.025 | 0.03 | |
| 1 | 1.02500 | 1.03000 |
| 2 | 1.05063 | 1.06090 |
| 3 | 1.07689 | 1.09273 |
The numbers in the body of the table, called tabular values, depend on or are determined by the numbers along the edge of the table, called arguments. In this example, if 1, 2, 3 are choices of a number n, and if 0.025, 0.03 are choices of a number i, then each tabular value y is equal to 1 plus i raised to the nth power. n and i are also called independent variables, and y is called the dependent variable. The table expresses a function or formula or rule. The rule could be stated as: add i to 1; raise the result to the nth power.
constant
A number is said to be a constant if it has the same value under all conditions. For example, in the formula:
(area of a circle) = π × (radius)²,
π is a constant, equal to 3.14159 ...,
applying equally well to all circles.
infinity
Mathematics recognizes several kinds of infinity. One of them occurs when numbers become very large. For example, the quotient of 12 divided by a number x, as x becomes closer and closer to 0, becomes indefinitely large, and the limit is called infinity and is denoted ∞.
equation, simultaneous, linear
An example of two linear simultaneous equations is:
7x + 8t = 22
3x + 5t = 11
x and t are called unknowns—that is, unknown variables—because the objective of solving the equations is to find them. These equations are called simultaneous because they are to be solved together, at the same time, for values of x and t which will fit in both equations. The equations are called linear because the only powers of the unknowns that appear are the first power. Values that solve equations are said to satisfy them. It is easy to solve these two equations and find that x = 2 and t = 1 is their solution. But it is a long process to solve 10 linear simultaneous equations in 10 unknowns, and it is almost impossible (without using a mechanical brain) to solve 100 linear simultaneous equations in 100 unknowns.
derivative, integral,
differential equation, etc.
See the sections in [Chapter 5] entitled “Differential Equations,” “Physical Problems,” and “Solving Physical Problems.” There these ideas and, to some extent, also the following ideas were explained: formula, equation, function, differential function, instantaneous rate of change, interval, inverse, integrating. See also a textbook on calculus. If y is a function of x, then a mathematical symbol for the derivative of y with respect to x is Dₓ y, and a symbol for the integral of y with respect to x, is ∫y dx. An integral with given initial conditions ([see p. 83]) is a definite integral.
exponential
A famous mathematical function is the exponential. It equals the constant e raised to the x power, eˣ, where e equals 2.71828.... The exponential lies between the powers of 2 and the powers of 3. It can be computed from:
| eˣ = 1 + | x² | + | x³ | + . . . |
| 1 · 2 | 1 · 2 · 3 |
It is a solution of the differential equation Dₓy = y. See also a textbook on calculus. The exponential to the base 10 is 10ˣ.
logarithm
Another important mathematical function is the logarithm. It is written log x or logₑ x and can be computed from the two equations:
log uv = log u + log v
| log(1 + x) = x - | x² | + | x³ | - . . . x² < 1 |
| 2 | 3 |
It is a solution of the differential equation Dₓy = 1/y. If y is the logarithm of x, then x is the antilogarithm of y. The logarithm to the base 10 of x, log₁₀ x, equals the logarithm to the base e of x, logₑ x, divided by logₑ 10. See also textbooks on algebra and calculus.
sine, cosine, tangent, antitangent
These also are important mathematical functions. The sine and cosine are solutions of the differential equation Dₓ(Dₓy) =-y and are written as sin x and cos x. They can be computed from
| sin x = x - | x³ | + | x⁵ | - . . . |
| 1 · 2· 3 | 1 · 2 · 3· 4· 5 | |||
| cos x = 1 - | x² | + | x₄ | - . . . |
| 1 · 2 | 1 · 2 · 3· 4 | |||
The tangent of x is simply sine of x divided by cosine of x. If y is the tangent of x, then x is the antitangent of y. See also references on trigonometry and on calculus. Trigonometric tables include sine, cosine, tangent, and related functions.
Bessel functions
These are mathematical functions that were named after Friedrich W. Bessel, a Prussian astronomer who lived from 1784 to 1846. Bessel functions are found as some of the solutions of the differential equation
x² Dₓ(Dₓy) + x Dₓy + (x² - n²)y = O
This equation arises in a number of physical problems in the fields of electricity, sound, heat flow, air flow, etc.
matrix
A matrix is a table (or array) of numbers in rows and columns, for which addition, multiplication, etc., with similar tables is specially defined. For example, the matrix
| 1 | 2 | |
| 3 | 4 |
plus the matrix
| 5 | 20 | |
| 60 | 100 |
equals the matrix
| 6 | 22 | |
| 63 | 104 |
(Can you guess the rule defining addition?)
Calculations using matrices are useful in physics, engineering, psychology, statistics, etc. To add a square matrix of 100 terms in an array of 10 columns and 10 rows to another such matrix, 100 ordinary additions of numbers are needed. To multiply one such matrix by another, 1000 ordinary multiplications and 900 ordinary additions are needed. See references on matrix algebra and matrix calculus.
differences, smoothness, checking
On [p. 221], a sequence of values of y is shown: 26, 37, 50, 65, 82. Suppose, however, the second value of y was reported as 47 instead of 37. Then the differences of y as we pass down the sequence would not be 11, 13, 15, 17 (which is certainly regular or smooth) but 21, 3, 15, 17 (which is certainly not smooth). The second set of differences would strongly suggest a mistake in the reporting of y. The smoothness of differences is often a useful check on a sequence of reported values.
Supplement 3
REFERENCES
A book like the present one can cover only a part of the subject of machines that think. To obtain more information about these machines and other topics to which they are related there are many references that may be consulted. There are still few books directly on the subject of machines that think, but there are many articles and papers, most of them rather specialized.
The purpose of this supplement is to give a number of these references and to provide a brief, general introduction to some of them. The references are subdivided into groups, each dealing with a branch of the subject. The references in each group are in alphabetical order by name of author (with “anonymous” last), and under each author they are in chronological order by publication date. Some publications, especially a forum or symposium, are listed more than once, according as the topic discussed falls in different groups. In this supplement, the sign three dots ( ...) next to the page numbers for an article indicates that the article is continued on later, nonconsecutive pages.
It seemed undesirable to try to make the group of references dealing with a subject absolutely complete, so long as enough were given to provide a good introduction to the subject. It proved impractical to try to make the citation of every single reference technically complete, so long as enough citation was given so that the reference could certainly be found. Furthermore, in a list of more than 250 references, errors are almost certain to occur. If any reader should send me additions or corrections, I shall be more than grateful.