Problem 1 - Calculus: One and Several Variables

Question

Is the number rational or irrational? \(\frac{17}{7}\).

Step-by-Step Solution

Answer

The number \(\frac{17}{7}\) is a rational number.

Step 1: Identify the number

Here, the given number is \(\frac{17}{7}\).

Step 2: Analyse whether it can be expressed as a fraction

The number is already in a fraction format. It's in the form of \(\frac{a}{b}\) where 'a' and 'b' are integers and 'b' is not equal to zero.

Step 3: Conclusion

Since the number is in the form of a fraction, it is a rational number.

Key Concepts

Fractions

Understanding the concept of fractions is crucial when dealing with rational numbers. A fraction is a way to represent a part of a whole. It is written in the form \(\frac{a}{b}\), where 'a' is the numerator, and 'b' is the denominator. The denominator shows into how many parts the whole is divided, while the numerator indicates how many of those parts are considered.

Examples of fractions:

• \(\frac{1}{2}\) represents one out of two equal parts.
• \(\frac{3}{4}\) signifies three out of four equal segments.
• \(\frac{17}{7}\) means seventeen parts of a whole that is divided into seven.

A fraction can be either proper, improper, or mixed.

• A proper fraction has a numerator smaller than the denominator, like \(\frac{1}{3}\).
• An improper fraction has a numerator higher than or equal to the denominator, such as \(\frac{7}{5}\).
• A mixed number combines a whole number and a fraction, like \(1\frac{1}{4}\).

Understanding these variations helps in recognizing rational numbers efficiently.

Rational and Irrational Numbers

Numbers can be categorized as either rational or irrational. A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where 'a' and 'b' are integers and 'b' is not zero. This means rational numbers include fractions, decimals that terminate or repeat, and whole numbers.

• Examples of rational numbers: \(\frac{1}{2}, \frac{17}{7}, 0.75,\) and 8.

Irrational numbers, on the other hand, cannot be represented as a simple fraction. They are non-repeating, non-terminating decimals like \(\pi\) and \(\sqrt{2}\). These numbers go on forever without any repeated pattern.
Understanding the difference between these types of numbers helps in solving mathematical problems and recognizing number patterns. For instance, since \(\frac{17}{7}\) is expressed as a fraction, it confirms that it's a rational number.

Integer Division

When dealing with fractions, it's often necessary to perform integer division. This process involves dividing one integer by another. The quotient may be a whole number or a decimal that can be expressed as a fraction. For instance, dividing 17 by 7 using integer division yields the fraction \(\frac{17}{7}\).

• If the result of the division is a whole number, such as 20 divided by 5, the result is 4, which is also rational.
• If the division leads to a fraction, like 17 divided by 7, it further supports that 17 divided by 7 remains a rational number.
The concept of integer division is vital in simplifying arithmetic problems and understanding how whole numbers relate to fractions. By repeating such division, you can also unfold the decimal representation of a fraction, which sometimes reveals its rational nature.