Operations with Rational and Irrational Numbers When performing operations with rational and irrational numbers, there are some rules and facts to consider: ● The sum (or difference) of any two rational numbers is rational. Examples: 5 + 18 = 23 2 - 3 = -1 2.3 + 6 = 8.3 3/10 + 2/10 = 5/10 = ½ Other examples? ● The sum (or difference) of any rational number and any irrational number is irrational. Examples: 5 + π = 5 + 3.141592…. = 8.141592... √4 + √2 = 2 + √2 = 2+ 1.414213… = 3.414213… π - 2 = 3.141592… - 2 = 1.141592… Other examples? ● The product of any two rational numbers is rational. Examples: 2 x 6 = 12 2.5 x 3 = 7.5 ⅔ x ⅓ = 2/9 Other examples? ● The product of any non-zero rational number and any irrational number is irrational. Examples: 5 x π = 5 x 3.141592…. = 15.70796… √2 x 3 = 1.414213… x 3 = 4.24264… Other examples? Note: The only time the product of a rational and an irrational results in a rational number is when the rational factor is zero. Example: 0 x √2 = 0 which is a rational number Caveat: The sum of two irrational numbers might be rational or irrational. Examples: √2 + √2 = 1.41213… + 1.41213… = 2.82842… (√2 + 2) + (5 - √2) = √2 - √2 + 2 + 5 = 7 Likewise, the product of two irrational numbers might be rational or irrational. Examples: √2 x √3 = 1.41213… x 1.76205… = 2.449489… √2 x √2 = 1.41213...x 1.41213… = 2 Each of these have to be looked at on a case by case basis