Actually here _y_ = _x_² + 1, and so the true value of _y_₇ˌ₂ is (7.2 × 7.2) + 1, or 52.84, which is rather close to 53. Types of interpolation procedures more accurate than linear interpolation will come much nearer still to the true value. 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