Problem 51 - Calculus: One and Several Variables

Question

Show that if \(0 \leq a \leq b\), then \(\sqrt{a} \leq \sqrt{b}\).

Step-by-Step Solution

Answer

The proof shows that if a and b are real numbers such that 0 ≤ a ≤ b, then it implies that √a ≤ √b. This is due to the SQRT-preserving property of inequalities which states that the square root function preserves the order of real numbers.

Step 1: Assumption

We start by assuming that a and b are real numbers such that 0 ≤ a ≤ b. This means that a is less than or equal to b.

Step 2: Square the inequalities

To demonstrate this principle we will square both sides of the inequality. Since a and b are both non-negative, we know that squaring them will not change the direction of the inequality. Squaring the given inequality, a ≤ b becomes, a^2 ≤ b^2

Step 3: Apply the Square Root

Next, apply square root on both sides of the inequality and considering that the square root function is increasing for non-negative arguments. This allows us to apply the square root to both sides of the inequality without altering the inequality direction. Thus, √a^2 ≤ √b^2.

Step 4: Simplify

Squaring and then square rooting a number gives back the absolute value of the original number. As we started with non-negative numbers, we could simplify √a^2 to a and √b^2 to b which yields the inequality a ≤ b.

Step 5: Conclusion

Thus, if a and b are real numbers such that 0 ≤ a ≤ b then √a ≤ √b holds. It shows that the square root function preserves the order of real numbers.

Key Concepts

Square root function

The square root function is fundamental in mathematics. It operates by reversing the process of squaring a number. This function particularly acts on non-negative numbers, as the square root of negative numbers isn't defined within real numbers. To give a precise example, the square root of 9 is 3, because 3 squared (or multiplied by itself) is 9. The square root function is denoted with the symbol \( \sqrt{ } \), such that \( \sqrt{x} \cdot \sqrt{x} = x \).

An important characteristic of the square root function is that it is a monotonic increasing function when restricted to non-negative values. This means that if you have two non-negative numbers and one is larger than the other, the square roots of these numbers will preserve this order. This behavior is crucial for solving inequalities involving square roots.

Non-negative numbers

Non-negative numbers are an essential part of understanding square roots and inequalities. These numbers are either greater than or equal to zero. They include all positive real numbers and zero, and can be visualized on a number line as starting from zero and extending to positive infinity.

In mathematical inequalities, a common requirement is for the numbers to be non-negative, especially when dealing with functions like the square root. The necessity stems from the fact that operations involving non-negative numbers are straightforward, particularly in terms of maintaining the direction of inequalities when taking roots or expanding with powers.

Order preservation

Order preservation is a fashionable trait in mathematics, particularly regarding functions. A function is said to preserve order if whenever two numbers are ordered in a certain way, their function values maintain the same order. In context, let’s consider the square root function that maintains order for non-negative numbers.

This means for any two non-negative numbers \( a \) and \( b \), if \( a \leq b \), then \( \sqrt{a} \leq \sqrt{b} \). This is because the square root function does not reverse or change the order of values that are under its operation. This property is utilized extensively when simplifying or solving inequalities. It allows confidence in manipulating inequalities involving the square root function without fearing a change in inequality direction.

Real numbers

Real numbers form the backbone of everyday mathematics, representing a continuum of numbers that include both rational and irrational numbers. Real numbers can be positive, negative, or zero, encompassing a vast range of magnitudes and decimal expansions. Examples include integers like 3, fractions like \( \frac{1}{2} \), and irrational numbers like \( \sqrt{2} \).

In the problem of inequalities involving square roots, we focus on the non-negative subset of real numbers. This is pertinent because the square root of a real number is only defined when the number is non-negative. When examining inequalities in real numbers, attention to whether we're dealing with positive, negative, or zero values is crucial, as operations may differ or be undefined outside certain domains.