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Introducing interval notation in VS .NET
Introducing interval notation
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You can alleviate the awkwardness of writing answers with inequality notation by using another format called interval notation You use interval notation extensively in calculus, where you re constantly looking at different intervals involving the same function Much of higher mathematics uses interval notation, although I really suspect that book publishers pushed its use because it s quicker and neater than inequality notation Interval notation uses parentheses, brackets, commas, and the infinity symbol to bring clarity to the murky inequality waters
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Part I: Homing in on Basic Solutions
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And, surprise surprise, the interval-notation system has some rules: You order any numbers used in the notation with the smaller number to the left of the larger number You indicate or equal to by using a bracket If the solution doesn t include the end number, you use a parenthesis When the interval doesn t end (it goes up to positive infinity or down to negative infinity), use + or , whichever is appropriate, and a parenthesis Here are some examples of inequality notation and the corresponding interval notation: x < 3 ( , 3) x 2 [ 2, ) 4 x < 9 [4, 9) 3 < x < 7 ( 3, 7) Notice that the second example has a bracket by the 2, because the greater than or equal to indicates that you include the 2, also The same is true of the 4 in the third example The last example shows you why interval notation can be a problem at times Taken out of context, how do you know if ( 3, 7) represents the interval containing all the numbers between 3 and 7 or if it represents the point ( 3, 7) on the coordinate plane You can t tell A problem containing such notation has to give you some sort of hint
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Compounding inequality issues
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A compound inequality is an inequality with more than one comparison or inequality symbol for instance, 2 < x 5 To solve compound inequalities for the value of the variables, you use the same inequality rules (see the intro to this section), and you expand the rules to apply to each section (intervals separated by inequality symbols) To solve the inequality 8 3x 5 < 10, for example, you add 5 to each of the three sections and then divide each section by 3: - 8 # 3x - 5 < 10 +5 +5 +5 - 3 # 3x < 15 - 3 # 3x < 15 3 3 3 -1 # x <5
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2: Toeing the Straight Line: Linear Equations
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Ancient symbols for timeless operations
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Many ancient cultures used their own symbols for mathematical operations, and the cultures that followed altered or modernized the symbols for their own use You can see one of the first symbols used for addition in the following figure, located on the far left a version of the Italian capital P for the word piu, meaning plus Tartaglia, a self-taught 16th century Italian mathematician, used this symbol for addition regularly The modern plus symbol, +, is probably a shortened form of the Latin word et, meaning and Greek times for subtraction The modern subtraction symbol, , may be a leftover from what the traders in medieval times used to indicate differences in product weights Leibniz, a child prodigy from the 17th century who taught himself Latin, preferred the third symbol from the left for multiplication One modern multiplication symbol, x or , is based on St Andrew s Cross, but Leibniz used the open circle because he thought that the modern symbol looked too much like the unknown x The symbol on the far right is a somewhat backward D, used in the 18th century by French mathematician Gallimard for division The modern division symbol, , may come from a fraction line with dots added above and below
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