Algebra for prep 2

🤔 Think and Discuss: Properties of Real Numbers

An analysis of fundamental properties for any real numbers \(a, b, c\).

1. Is \(a - b = b - a\)?

This question asks if subtraction is commutative. The commutative property states that you can swap the order of the numbers without changing the result.

Let's test this with a simple example. Let \(a = 5\) and \(b = 3\).

• \(a - b = 5 - 3 = 2\)
• \(b - a = 3 - 5 = -2\)

Since \(2 \neq -2\), the statement is not true in general.

Answer: False. Subtraction is not commutative.

2. Is \(\frac{a}{b} = \frac{b}{a}\)?

This question asks if division is commutative. Let's test this with an example. Let \(a = 6\) and \(b = 3\).

• \(\frac{a}{b} = \frac{6}{3} = 2\)
• \(\frac{b}{a} = \frac{3}{6} = 0.5\)

Since \(2 \neq 0.5\), the statement is not true in general. (Note: This only holds if \(a = b\) or \(a = -b\), assuming \(a, b \neq 0\)).

Answer: False. Division is not commutative.

3. Is \(\frac{a}{b} \in \mathbb{R}\)? (Is the result a real number?)

This question asks if the result of dividing any two real numbers is always another real number. The set of real numbers is denoted by \(\mathbb{R}\).

This is true for almost all cases. However, there is one critical exception: division by zero. In mathematics, dividing any number by zero is undefined.

If we let \(b = 0\), then the expression \(\frac{a}{0}\) is not a real number.

Answer: Not always. It is only a real number if the denominator \(b \neq 0\).

4. Is \(a(b - c) = ab - ac\)?

This question describes the distributive property of multiplication over subtraction. This property states that multiplying a number by a group of numbers added or subtracted together is the same as doing each multiplication separately.

This is a fundamental, axiomatic property of real numbers. It is always true.

Answer: True. This is the distributive property.