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