6.1 Roots and Sets

When simplified or evaluated, a subexpression like √9 is transformed to a positive number if the root is an integer but note that information is lost because the negative root is discarded. If it is necessary to retain both roots, special simplification of √9 transforms to ±3. But it is important to realize that the plus-or-minus unary operator really evaluates to two values (unless the operand is zero). To work with expressions like this, the distribution of unary ± produces a set with two elements. That is,

Note this will very likely change the type of the expression in which it appears.

A real example of this happens when complete-the-square (§3.4.10) is applied to

After simplifying and isolating x-1, the equation appears as x-1=±2. This is where a set is introduced. Distributing x-1=.{±2} produces .[x-1={2, -2}] (note this expression is invalid because it is unbalanced with respect to type). After isolating x, the expression becomes xʂ={2, -2}+1 and simplifying again produces xʂ={3, -1}.

Binary ± operates in a similar way. An example is this: completing the square of .{3⋅x^2+2⋅x-1}=0 leads to 3⋅(x+1/3)^2-4/3=0; isolating x gives x=±(√(4/9))-1/3. Now, commute the the term with the radical to the right and distribute the radical: x=-(1/3)+±(2/3) (here, ± is a unary operator); and simplify again: x=-(1/3)±2/3 (here, ± is a binary operator). Distribute the binary ± and simplify to give the roots xʂ={1/3, -1}.

Working with unary and binary ± leads to non-scalar intermediate results that can lead, in turn, back to scalar expressions. The remainder of this section explores this idea.

Distributing unary ± produces a set in which the operand of ± appears twice, once negated: ±3 Distribute is {3, -3}.

Applying Factor to a set consisting of two elements, one of which is the negation of the other, produces a unary ± whose operand is the square root of the positive set element, squared.

Distribute, factor and simplify provide a three-cornered cycle for unary ±. Try this: enter the four-part equivalence ±3=±3=±3=±3. Distribute the second part; distribute and factor the third part; for the fourth part, distribute and factor it, then evaluate its radical. You will end up with the following equivalence: .[±3={3, -3}=√(3^2)=±3].

Distributing binary ± produces a set in which the left operand is combined with the right operand, once using addition and once using subtraction: .{2±3} Distribute is {2+3, 2-3}.

Factoring a set consisting of two elements, one of which is an addition and the other a subtraction with matching corresponding operands, produces a binary ± whose operands are those of either of the elements. That is, .{{2+3, 2-3}} Factor is 2±3.