Problem 8 - Intermediate Algebra

Question

Identify each statement as true or false. Zero is a rational number.

Step-by-Step Solution

Answer

True, zero is a rational number.

Step 1: Understanding Rational Numbers

A rational number is defined as a number that can be expressed as the quotient or fraction of two integers \( p/q \), where \( q eq 0 \). Both \( p \) and \( q \) must be integers.

Step 2: Expressing Zero as a Fraction

Zero can be expressed as a fraction \( 0/1 \), where 0 is the numerator and 1 is the denominator. Since 1 is a non-zero integer, 0 fits the definition of a rational number.

Step 3: Conclusion

Since zero can be written as a fraction \( 0/1 \), which meets the criteria for being a rational number, the statement is true.

Key Concepts

Integers

Integers are the set of whole numbers that include negative numbers, zero, and positive numbers. They are represented by the symbols

• ..., -3, -2, -1, 0, 1, 2, 3, ...

This sequence clearly shows integers are continuous and extend infinitely in both the positive and negative directions. They do not include fractions or decimals, merely whole numbers.
Integers are fundamental in mathematics because they are used to construct other number sets like rational numbers. Understanding integers helps to grasp how rational numbers function since any integer can be expressed as a ratio of integers by using 1 as the denominator. For instance, the integer 5 can be expressed as a rational number in the form of \[\frac{5}{1}\].

Fractions

A fraction represents a part of a whole and is written as \[\frac{p}{q}\] where \(p\) is the numerator, representing the number of parts we have, and \(q\) is the denominator, representing the number of equal parts the whole is divided into.Fractions are closely related to rational numbers, as

• Every fraction is a rational number.
• The denominator \(q\) in a fraction can never be zero.

The concept of fractions allows us to compare quantities and parts of a whole, enabling various calculations in mathematics. They make it possible to express quantities in forms that are precise and easy to work with in algebraic expressions.

Zero as a Rational Number

Zero is often a source of confusion when discussing rational numbers, but it is indeed one of them. A rational number is expressed as \[\frac{p}{q}\] where both \(p\) and \(q\) are integers, and \(q\) is non-zero. Zero can be expressed as the fraction \[\frac{0}{1}\], with 0 as the numerator, and 1 as the denominator, which fits the definition.

• The numerator can be zero, which makes the whole value of the fraction zero.
• The denominator should be a non-zero integer. In this case, it's 1.

Thus, zero is a rational number because it can be expressed as the ratio of two integers, satisfying the condition that the denominator is not zero.