Problem 55 - Prealgebra

Question

OPEN ENDED Give an example of a number that is an integer and a rational number.

Step-by-Step Solution

Answer

7 is both an integer and a rational number.

Step 1: Define Integers

Integers are whole numbers that can be positive, negative, or zero. Examples include -3, 0, 1, and 42.

Step 2: Define Rational Numbers

Rational numbers are numbers that can be expressed as the fraction of two integers (a/b) where 'b' is not zero. For example, 1/2, -5/3, and 7 are rational numbers.

Step 3: Identify Overlap

To find a number that is both an integer and a rational number, we need a number that fits both definitions. Since an integer like 7 can be expressed as a fraction like 7/1, it is also a rational number.

Key Concepts

Rational Numbers

Rational numbers are an important concept in mathematics. They include any number that can be expressed as a ratio of two integers, meaning one integer divided by another. By definition, the denominator cannot be zero, as division by zero is undefined. For example, the number \( \frac{3}{4} \) is a rational number because both 3 and 4 are integers, and 4 is not zero. Actually, any integer is also a rational number. For instance, the integer 5 can be written as \( \frac{5}{1} \), satisfying the condition of a rational number.

• Includes fractions and integers
• Can be positive or negative
• The denominator (the bottom number in the ratio) cannot be zero

Understanding rational numbers is key when dealing with mathematical equations and real-world problems, because they describe quantities that are not whole units.

Whole Numbers

Whole numbers are the set of numbers that include all of the non-negative integers starting from zero. In simpler terms, whole numbers are zero and all the positive numbers without fractions or decimals. This set includes numbers like 0, 1, 2, 3, and so forth. They do not include negative numbers, fractions, or decimals.

• Start from 0 and go upwards
• No fractions or decimals
• Always non-negative

Each whole number is a complete unit, making them easy to work with and understand. For example, when counting objects, you naturally think in whole numbers, as there is no 'half' or 'quarter' included. Understanding whole numbers lays the foundation for more complex math concepts, like integers and rational numbers.

Fractions

Fractions are a way to represent a part of a whole. They consist of two numbers separated by a slash: the numerator (top number) and the denominator (bottom number). The numerator indicates how many parts you have, while the denominator indicates the total number of parts that make up a whole. For instance, in the fraction \( \frac{1}{2} \), 1 is the numerator, and 2 is the denominator, signifying that you have one part of a whole divided into two parts.

• Represents parts of a whole
• Has a numerator and a denominator
• Can be converted to decimals and percentages

Fractions are especially useful in real-life situations where you might not be dealing with whole items, such as when splitting a pizza among friends. They help in widening our understanding beyond whole numbers and are crucial for working with rational numbers.