When it comes to crossword puzzles, clues can be a bit tricky and challenging to decipher. One such clue is “Figure whose squares are positive.” It may sound complicated and daunting, especially for those who are not well-versed in mathematics. However, the answer to this clue is relatively simple, and with a bit of understanding of mathematical concepts, anyone can solve it.

Here, we will discuss the mathematical concept behind the “figure whose squares are positive crossword clue, explain what it means, and provide some examples to help you understand it better.

Understanding the Concept of Squares

Before we delve deeper into the meaning of the clue, let us first understand the concept of squares in mathematics. In simple terms, the square of a number is the number multiplied by itself. For example, the square of 3 is 3 x 3 = 9. Similarly, the square of -3 is -3 x -3 = 9.

Now, when we talk about the squares of numbers, we can classify them into two categories: positive squares and negative squares. Positive squares are the squares of positive numbers, while negative squares are the squares of negative numbers.

For instance, 9 is a positive square because it is the square of a positive number (3). On the other hand, -9 is a negative square because it is the square of a negative number (-3).

Understanding the Clue: “Figure whose squares are positive”

Now that we have a basic understanding of the concept of squares let us move on to the clue “figure whose squares are positive.”

In mathematical terms, this clue is referring to a figure or a number whose square is always positive, no matter what. In other words, it is a number that is never negative when squared.

So, what numbers can be classified as “figure whose squares are positive”? The answer is quite simple. All positive numbers, including 0, can be classified as “figure whose squares are positive.”

For example, the square of 5 is 25, which is a positive number. Similarly, the square of 0 is 0, which is also a positive number.

However, negative numbers and imaginary numbers cannot be classified as “figure whose squares are positive.” The square of a negative number is always positive, but the negative sign makes it negative. For example, the square of -5 is 25, which is a positive number, but the negative sign makes it negative.

Examples of “Figure whose squares are positive” Crossword Clues

Now that we have a better understanding of the mathematical concept behind the clue, let us look at some examples of “figure whose squares are positive” crossword clues.

1. The first example is “Square root of 16, whose squares are positive.” The answer to this clue is 4. The square of 4 is 16, which is a positive number.
2. Another example is “Perfect square whose squares are positive.” The answer to this clue is any positive integer that is the square of another integer, such as 1, 4, 9, 16, and so on.
3. “Square of a real number, whose squares are positive” is another example. The answer to this clue is any positive real number, including 0.

Another interesting property of positive definite matrices is their relation to the concept of an ellipsoid. An ellipsoid is a three-dimensional shape that can be thought of as a stretched or compressed sphere. More precisely, an ellipsoid is defined as the set of all points in three-dimensional space that satisfy the equation

(x – x0)T A (x – x0) = 1

where x0 is the center of the ellipsoid, A is a positive definite matrix, and (x – x0)T denotes the transpose of the vector (x – x0).

The reason this equation defines an ellipsoid is that it describes all points in space that are a fixed distance from the center of the ellipsoid, where the distance is measured using the matrix A. In other words, if we take any point x on the surface of the ellipsoid, the vector (x – x0) will be tangent to the surface at x, and the matrix A will determine the curvature of the surface at x.

The properties of positive definite matrices have many applications in various fields of mathematics and science, including optimization, statistics, and physics. For example, positive definite matrices play a central role in the theory of least-squares regression, where the goal is to find the best-fit line or curve for a given set of data. In this context, the positive definite matrix is used to measure the quality of the fit, and the least-squares solution can be obtained by minimizing a quadratic form involving the matrix.

Another important application of positive definite matrices is in the study of partial differential equations, where they arise as the coefficient matrices for various types of boundary value problems. Positive definite matrices are also used in numerical analysis, where they are used to construct efficient and stable algorithms for solving systems of linear equations.

In summary, a positive definite matrix is a square matrix that has many important properties and applications in mathematics and science. These matrices are used to define inner products, measure distances, and characterize the curvature of surfaces. Positive definite matrices are also used in a variety of applications, including optimization, statistics, physics, and numerical analysis. Understanding the properties of positive definite matrices is essential for anyone studying these fields, and can lead to many insights and discoveries.