Which of the following statements is mathematically false?

The correct option (the statement that is mathematically false) is: Every real number is an irrational number.

Step-by-step Explanation:

1. Statement 1: "Every real number is an irrational number."
The set of real numbers consists of both rational numbers and irrational numbers.
Rational numbers are numbers that can be expressed as a fraction of two integers, such as 1, 2, 0.5, or 3/4.
Irrational numbers are numbers that cannot be expressed as a ratio of two integers, such as
$\sqrt{2}$
or
$\pi$
Because rational numbers are real numbers but are not irrational numbers, it is mathematically false to state that every real number is an irrational number.

2. Statement 2: "All natural numbers are real numbers."
Natural numbers (1, 2, 3, ...) are positive integers. Since all integers are part of the real number system, all natural numbers are real numbers. This statement is true.

3. Statement 3: "Every integer can be plotted as a point on the number line."
The real number line represents all real numbers. Since every integer is a real number, every integer corresponds to a specific point on the real number line. This statement is true.

4. Statement 4: "All integers are rational numbers."
Any integer
$n$
can be written in the rational form
$\frac{n}{1}$
where both numerator and denominator are integers and the denominator is not zero. Thus, all integers are rational numbers. This statement is true.

Conclusion:
Therefore, the statement "Every real number is an irrational number." is the only mathematically false statement.