# 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
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.

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.

### Relevant Questions

Discover prove: Combining Rational and Irrationalnumbers is $$\displaystyle{1.2}+\sqrt{{2}}$$ rational or irrational? Is $$\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}$$ rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.
The rational numbers are dense in $$\displaystyle\mathbb{R}$$. This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in $$\displaystyle\mathbb{R}$$ as well.
True or False?
1) Let x and y real numbers. If $$\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}$$ and $$\displaystyle{x}\ne{y}$$, then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then $$\displaystyle{x}^{{2}}$$ is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
Find
a) a rational number and
b) a irrational number between the given pair.
$$\displaystyle{3}\frac{{1}}{{7}}$$ and $$\displaystyle{3}\frac{{1}}{{6}}$$
Determine whether the below given statement is true or false. If the statement is false, make the necessary changes to produce a true statement:
All irrational numbers satisfy |x - 4| > 0.
In which set(s) of numbers would you find the number $$\displaystyle\sqrt{{80}}$$
$$\frac{1}{4}$$ and $$\frac{3}{4}$$