No.
3*sqrt(2) and sqrt(2) are irrational. But their quotient is 3, which is rational.
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Yes, an irrational number divided by an irrational number will usually be an irrational number, but not always. For example, pi / (3pi) is 1/3, which is a rational number, but pi/e is an irrational number.
You can add any irrational number.
0.4444
Any irrational number, when multiplied by 0.5 will give an irrational number.
Any irrational number, added to 0.4 will give an irrational number.