Consider the following statements. Select all that are always true. The sum of a rational number and a rational number is rational. The sum of a rational number and an irrational number is irrational. The sum of an irrational number and an irrational number is irrational. The product of a rational number and a rational number is rational. The product of a rational number and an irrational number is irrational. The product of an irrational number and an irrational number is irrational.

Question
Irrational numbers
asked 2021-02-11
Consider the following statements. Select all that are always true.
The sum of a rational number and a rational number is rational.
The sum of a rational number and an irrational number is irrational.
The sum of an irrational number and an irrational number is irrational.
The product of a rational number and a rational number is rational.
The product of a rational number and an irrational number is irrational.
The product of an irrational number and an irrational number is irrational.

Answers (1)

2021-02-12
1.Statement: The sum of a rational number and a rational number is rational. The sum of a rational number and a rational number is every time rational. For example,
\(\displaystyle\frac{{1}}{{2}}+\frac{{2}}{{1}}=\frac{{{1}+{4}}}{{{2}\cdot{1}}}=\frac{{5}}{{4}}\)
Thus, the given statement is always true.
2.Statement: The sum of a rational number and an irrational number is irrational. The sum of a rational number and an irrational number is every time irrational.т For example,
\(\displaystyle\frac{{1}}{{2}}+{\left(\frac{{1}}{{2}}+\sqrt{{2}}\right)}={1}+\sqrt{{2}}\) (irrational)
Thus, the given statement is always true.
3.Statement: The sum of an irrational number and an irrational number is irrational. тThe sum of an irrational number and an irrational number is sometimes rational. For example,
\(\displaystyle{\left(\frac{{1}}{{2}}+\sqrt{{2}}\right)}+\sqrt{{2}}=\frac{{1}}{{2}}+{2}\sqrt{{2}}\) (irrational)
and
\(\displaystyle{\left(\frac{{1}}{{2}}+\sqrt{{2}}\right)}+{\left(\frac{{1}}{{2}}-\sqrt{{2}}\right)}=\frac{{1}}{{2}}+\frac{{1}}{{2}}+\sqrt{{2}}-\sqrt{{2}}={1}+{0}={1}\) (rational)
Thus, the given statement is false.
4. Statement: The product of a rational number and a rational number is rational. The product of two rational numbers is rational every time. For example,
\(\displaystyle\frac{{1}}{{2}}\cdot\frac{{2}}{{1}}=\frac{{2}}{{2}}\) (rational)
Thus, the statement is always true.
5.Statement: The product of a rational number and an irrational number is irrational.The product of a rational number and an irrational number is every time irrational. For example,
\(\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}=\frac{\sqrt{{2}}}{{2}}=\frac{{1}}{\sqrt{{2}}}\) (irraational)
Thus, the given statement is always true.
6.Statement: The product of an irrational number and an irrational number is irrational. The product of an irrational number and an irrational number is sometimes rational. For example,
\(\displaystyle\sqrt{{2}}\cdot\sqrt{{3}}=\sqrt{{6}}\) (irrarional)
and
\(\displaystyle\sqrt{{3}}\cdot\sqrt{{27}}=\sqrt{{{3}\cdot{27}}}=\sqrt{{81}}={9}\) (rational)
Thus, the statement is false.
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