Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Indicate whether the statement is true or false. Every integer is a rational number.

Step-by-Step Solution

Answer

True. Every integer can be expressed as a rational number with a denominator of 1, such as \(a/1\), where "a" is an integer.

Step 1: Understand the definitions of integers and rational numbers

An integer is a whole number, which can be positive, negative, or zero (e.g. -3, -2, -1, 0, 1, 2, 3). Rational numbers are numbers that can be expressed as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers and the denominator is not equal to zero.

Step 2: Analyze the relationship between integers and rational numbers

Consider an integer, let's call it "a". We can express "a" as a fraction by using "1" as the denominator, like this: \(a/1\). Since both "a" and "1" are integers, and the denominator is not zero, this fraction is a rational number.

Step 3: Conclusion

Based on our analysis, we can now answer the given statement. Every integer can be expressed as a rational number (with a denominator of 1), so the statement is true.

Key Concepts

Integers

Integers are one of the fundamental building blocks of mathematics. They encompass all the whole numbers that are either positive, negative, or zero. Here are some key points about integers:

• Whole numbers: These are numbers without fractions or decimals. Examples include \(-3, 0, 7\).
• Part of the number line: The number line extends infinitely in both directions, with integers scattered evenly along it.
• No fractions or decimals: An integer will never have a fractional or decimal part, making them straightforward and easy to understand.

Every integer can be depicted in its pure form, such as 5 or -12, making them distinct from numbers like 1.5 or \(-3.2\), which are not integers. The simplicity of integers contributes to their role as a stepping stone to more complex mathematical concepts.

Fractions

Fractions represent parts of a whole and are crucial for understanding rational numbers. They consist of two components: the numerator and the denominator. Here’s a closer look:

• Numerator: The top part of the fraction, indicating how many parts are being considered.
• Denominator: The bottom part, showing into how many equal parts the whole is divided.

Fractions like \(\frac{3}{4}\) can be interpreted as 3 parts of 4, representing the idea of dividing a whole into equal parts. The concept of fractions helps us perform operations like division, understand ratios, and make sense of portions in real-life situations.
It's important to note that fractions can denote equivalent values, for example, \(\frac{1}{2}\) is equal to \(\frac{2}{4}\). Understanding fractions lays the foundation for comprehending how integers can be represented as rational numbers, such as by expressing the integer 3 as \(\frac{3}{1}\).

Mathematical Definitions

Understanding mathematical definitions is crucial for grasping more complex concepts. Definitions provide clarity and help avoid misunderstandings.Key Definitions:

• Integer: Whole numbers that can be positive, negative, or zero.
• Rational Number: Numbers that can be expressed as a fraction where both the numerator and the denominator are integers, with the denominator not being zero.

Mathematical definitions serve as the bedrock for problem-solving and theory comprehension in mathematics. By clearly defining and distinguishing between concepts like integers and rational numbers, we can better evaluate statements like "Every integer is a rational number," and understand why it holds true. The integer can be expressed as \(\frac{a}{1}\), with \(a\) as any integer, demonstrating its place within rational numbers.