Question

Select all statements that are true about the real number system.

1) Irrational numbers contain decimal values that neither terminate nor repeat in a pattern.
2) Natural numbers, or counting numbers, can be positive or negative.
3) The real number system is made of rational numbers and irrational numbers.
4) Rational numbers can be written as a ratio.
5) Integers are positive and negative numbers, including zero, but do not include decimal or fractional values.

Answer 1

The real number system is comprised of both rational and irrational numbers. Rational numbers can be expressed as ratios, and integers are whole numbers without fractional parts. The statement about natural numbers being positive or negative is incorrect; natural numbers are exclusively positive.

Step-by-step explanation:

The real number system includes various types of numbers, some of which have specific characteristics that define them. Here are the statements with explanations:

• Irrational numbers contain decimal values that neither terminate nor repeat in a pattern. This is true. Examples include π (pi) and √2 (the square root of 2).
• Natural numbers, or counting numbers, can be positive or negative. This statement is false; natural numbers are only positive numbers starting from 1 upwards (1, 2, 3, ...).
• The real number system is made of rational numbers and irrational numbers. This is true, as the real number system encompasses all numbers that can be found on the number line. Rational numbers and irrational numbers are the two main categories of real numbers. Rational numbers can be expressed as fractions, whereas irrational numbers cannot.
• Rational numbers can be written as a ratio of two integers, where the denominator is not zero - this is also true. Examples include 1/2, -3, and 0.75.
• Integers are positive and negative numbers, including zero, but do not include decimal or fractional values. This statement is true. Integers include ... -3, -2, -1, 0, 1, 2, 3, ... but not numbers like 1.5 or √4.

Exact numbers, like counting objects (4 goats, 12 inches in a foot) are considered to have infinitely many significant figures and are part of what we can count or define clearly. When dealing with real numbers, we use digits to represent these values, and the presence of a decimal point (even if not explicitly shown, as in 123 for one hundred and twenty-three) indicates more precise measurements or divisions, which are characteristic of rational numbers. However, in scientific notation, even very small or large numbers can be expressed in a succinct and manageable form, helping us in various mathematical operations.