Which set includes all rational and irrational numbers?

• A. Real numbers
• B. Natural numbers
• C. Rational numbers
• D. Whole numbers

Answer: (A)

Think about what numbers can appear on a number line. Real numbers cover everything that can be measured or counted on that line, including both those you can write as a fraction of integers and those you can’t.

Rational numbers are numbers that can be written as a ratio of integers, like 1/2, -4, or 7. They can be placed exactly as fractions or integers on the number line. Irrational numbers are numbers that cannot be written as a ratio of integers; they go on the line as non-repeating, non-terminating decimals, like pi or the square root of 2.

Since every rational number and every irrational number lies somewhere on the number line, the set that contains both is the real numbers. The other sets described—natural numbers (positive integers), whole numbers (nonnegative integers), or rational numbers only—do not include all irrationals or all real numbers.

So, all rational and irrational numbers together make up the real numbers.