The Explain command asks a large language model to explain something. The item to be explained can be a string, an algebraic expression, a relation such as an equation or inequality, or a list or set of expressions or relations.

After an explanation, you can ask for more detail by issuing the command $\mathrm{Explain}\left(\mathrm{more}\right)$. This keyword can be given as a string or a name, and you can equivalently use the arguments $\mathrm{further}$, $\mathrm{continue}$, $\mathrm{go\; on}$, or $\mathrm{elaborate}$. Note that $\mathrm{go\; on}$ needs to be quoted because it contains a space.

Maple can interface with OpenAI's GPT-4o, o1-mini, and o3-mini models. You can select these by using the options $\mathrm{model}=\mathrm{GPT4o}$; $\mathrm{model}=\mathrm{O1Mini}$; and $\mathrm{model}=\mathrm{O3Mini}$, respectively. By default, or when explicitly selected with the $\mathrm{model}=\mathrm{default}$ option, a Maplesoft server chooses an appropriate model. (At the time of release of Maple 2025, this was the o3-mini model.) Note that OpenAI may deprecate and disable models, so the set of models supported may change in the future.

Note: Large language models often generate inaccurate statements. Please keep this in mind: this is not technology for building a bridge with.

$\mathrm{with}\left(\mathrm{NaturalLanguage}\right)\:$

$\mathrm{Explain}\left(x^2+y^2=1\right)$

$''The\; equation\; x²\; +\; y²\; =\; 1\; is\; most\; commonly\; associated\; with\; the\; unit\; circle\; in\; the\; Cartesian\; coordinate\; system.\; It\; represents\; all\; the\; points\; (x,\; y)\; that\; are\; exactly\; one\; unit\; away\; from\; the\; origin\; (0,\; 0).\; This\; equation\; is\; fundamental\; in\; many\; areas\; of\; math,\; especially\; in\; trigonometry,\; where\; the\; unit\; circle\; helps\; define\; the\; sine\; and\; cosine\; functions.''$

$\mathrm{Explain}\left(\mathrm{more}\right)$

$The equation x² + y² = 1 represents a circle in the Cartesian coordinate plane. More specifically, it's the equation of the unit circle—a circle with a radius of 1 and centered at the origin (0, 0). Here are some extra details: 1. Geometry: Every point (x, y) that satisfies the equation is exactly one unit away from the origin. That’s why when you square the distances (x and y) and add them up, you get 1, which is the square of the radius. 2. Trigonometry: The unit circle plays a key role in trigonometry. If you consider an angle θ measured from the positive x-axis, you can define x = cos θ and y = sin θ. This correspondence is fundamental because it allows us to interpret cosine and sine as the x- and y-coordinates, respectively, of a point on the unit circle. 3. Applications: Beyond geometry and trigonometry, the unit circle is used in calculus, complex analysis (where it represents the set of all complex numbers of modulus 1), and various fields of science and engineering. It provides a simple setting to explore periodic functions, oscillations, and rotations. Overall, x² + y² = 1 is a foundational equation that serves as a building block for many areas of mathematics.$

$\mathrm{Explain}\left(\mathrm{\Re }\left(\exp \left(x\mathrm{I}+y\mathrm{I}\right)\right)=\mathrm{\Re }\left(\exp \left(x\mathrm{I}\right)\right)\mathrm{\Re }\left(\exp \left(y\mathrm{I}\right)\right)-\mathrm{\Im }\left(\exp \left(x\mathrm{I}\right)\right)\mathrm{\Im }\left(\exp \left(y\mathrm{I}\right)\right)\right)$

$The equation is essentially a rephrasing of the cosine addition formula using Euler's formula. Here’s the breakdown: 1. Euler's formula tells us that for any real number θ, e^(iθ) = cos(θ) + i sin(θ). 2. If we let θ = x or y, then:   e^(i*x) = cos(x) + i sin(x) and e^(i*y) = cos(y) + i sin(y). 3. The real part of e^(i*x) is cos(x) and the imaginary part is sin(x) (and similarly for e^(i*y)). 4. When you compute e^(i*(x+y)), by Euler’s formula you get:   e^(i*(x+y)) = cos(x+y) + i sin(x+y). Thus, its real part is cos(x+y). 5. On the other hand, multiplying e^(i*x) and e^(i*y) gives:   e^(i*x) * e^(i*y) = [cos(x)cos(y) − sin(x)sin(y)] + i[cos(x)sin(y) + sin(x)cos(y)]. The real part of this product is cos(x)cos(y) − sin(x)sin(y). 6. Therefore, equating the two expressions for the real part, we have:   cos(x+y) = cos(x)cos(y) − sin(x)sin(y). The equation Re(e^(i*x+i*y)) = Re(e^(i*x)) * Re(e^(i*y)) - Im(e^(i*x)) * Im(e^(i*y)) is just another way of expressing this identity. It highlights how trigonometric identities arise naturally from the properties of complex exponentials and Euler's formula.$

$\mathrm{Explain}\left(''go\; on''\,\mathrm{model}=\mathrm{O1Mini}\right)$

$The equation you've provided: \[ \text{Re}\left(e^{i x + i y}\right) = \text{Re}\left(e^{i x}\right) \cdot \text{Re}\left(e^{i y}\right) - \text{Im}\left(e^{i x}\right) \cdot \text{Im}\left(e^{i y}\right) \] is a representation that connects complex exponentials with trigonometric identities, specifically the **cosine addition formula**. Let's break down each component to understand what this equation is about and why it holds true. ### Breaking Down the Equation 1. **Complex Exponentials:** - Euler's formula states that for any real number \( \theta \): \[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \] - Here, \( e^{i x} \) and \( e^{i y} \) are complex numbers on the unit circle in the complex plane, corresponding to angles \( x \) and \( y \) respectively. 2. **Left Side of the Equation:** - \( e^{i x + i y} = e^{i(x + y)} \) - Applying Euler's formula: \[ e^{i(x + y)} = \cos(x + y) + i\sin(x + y) \] - Thus, the real part (Re) of \( e^{i(x + y)} \) is \( \cos(x + y) \): \[ \text{Re}\left(e^{i x + i y}\right) = \cos(x + y) \] 3. **Right Side of the Equation:** - Compute the real and imaginary parts of \( e^{i x} \) and \( e^{i y} \): \[ \text{Re}\left(e^{i x}\right) = \cos(x), \quad \text{Im}\left(e^{i x}\right) = \sin(x) \] \[ \text{Re}\left(e^{i y}\right) = \cos(y), \quad \text{Im}\left(e^{i y}\right) = \sin(y) \] - Plugging these into the right side: \[ \cos(x) \cdot \cos(y) - \sin(x) \cdot \sin(y) = \cos(x + y) \] (This is the **cosine addition formula**.) 4. **Connecting Both Sides:** - Both sides simplify to \( \cos(x + y) \), hence the equality holds: \[ \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \] ### Interpretation and Applications This equation essentially demonstrates how **complex numbers and their properties can be used to derive and understand trigonometric identities**. Here's what it's about: - **Complex Multiplication and Trigonometry:** The product of two complex exponentials \( e^{i x} \) and \( e^{i y} \) corresponds to adding their angles in the exponent. When you take the real part of this product, you're effectively retrieving the cosine of the sum of the two angles, which aligns with the cosine addition formula. - **Derivation of Trigonometric Identities:** By using Euler's formula and properties of complex numbers, you can derive various trigonometric identities in a more streamlined and sometimes more intuitive way compared to purely geometric or algebraic methods. - **Applications in Engineering and Physics:** This relationship is foundational in fields like signal processing, quantum mechanics, and electrical engineering, where complex exponentials are used to represent oscillations and waves. Understanding how their real and imaginary parts interact is crucial for analyzing wave interference, resonance phenomena, and more. ### Summary The given equation is a complex-analytic expression of the cosine addition formula. It showcases how the real part of the product of two complex exponentials corresponds to the cosine of the sum of their angles, bridging complex number theory with classical trigonometry. **In essence, it's about leveraging the properties of complex numbers to express and understand fundamental trigonometric relationships.** **Answer Summary:** It’s an expression of how the real part of the product of two complex exponentials equals cos(x + y). Specifically, using Euler’s formula, Re(e^{i(x+y)}) = Re(e^{ix})Re(e^{iy}) – Im(e^{ix})Im(e^{iy}) translates to the cosine addition formula. Thus, the equation illustrates the relationship between complex numbers and trigonometric identities by showing how cos(x + y) is built from cos x, cos y, sin x, and sin y.$

$\mathrm{Explain}\left(\mathrm{\Re }\left(\mathrm{\sigma }\right)<0\right)$

$\mathrm{Explain}\left(\mathrm{continue}\right)$

$\mathrm{Explain}\left(''the\; Maple\; command\; simplify''\right)$

$In Maple, the simplify command is used to take a given mathematical expression and reduce it to a “simpler” or more canonical form. In other words, it rewrites the expression so that redundant or unnecessarily complicated parts are removed, which can make further analysis or manipulation easier. Here are some key points about simplify in Maple: 1. Purpose:   • It combines like terms, reduces fractions, and performs algebraic manipulations to present the expression in a more “compact” form.   • It removes unnecessary parentheses, rewrites radicals, trigonometric expressions, logarithms, and other functions when possible. 2. Basic Usage:   • The command is typically used as simplify(expr) where expr is the expression you want to simplify.   • For example, if you enter simplify((x^2 - 1)/(x - 1)) for x ≠ 1, Maple may simplify it to x + 1, recognizing the factorization of x^2 − 1 into (x − 1)(x + 1). 3. Customization and Options:   • While the default simplification is often sufficient, Maple allows you to pass options or specify methods for particular kinds of simplification.   • This can be useful when you want to control whether, for example, algebraic, trigonometric, or symbolic assumptions are taken into account during the simplification process. 4. How It Works:   • Internally, simplify uses a variety of built-in rules and algorithms to determine which parts of the expression can be combined or rewritten.   • It is not always “smart” enough to reduce an expression to what a human might consider its simplest form, so sometimes further simplification or alternative techniques (like using factor, combine, or assuming certain variable properties) may be needed. 5. Practical Considerations:   • It is a great first step when trying to make sense of a complicated result from a computation.   • Always check that the simplified expression is valid under any specific assumptions you may have about the variables (for instance, domain restrictions or nonzero denominators). In summary, the Maple command simplify is a powerful tool for reducing expressions to a simpler form by applying a variety of algebraic and symbolic manipulation techniques, making it easier to work with and understand complex mathematical expressions in your Maple sessions.$

$\mathrm{Explain}\left(''Weight\; of\; a\; W10X54\; beam?''\right)$

$''A\; designation\; like\; "W10X54"\; is\; used\; in\; the\; American\; steel\; industry\; to\; describe\; a\; specific\; wide-flange\; (or\; “W”)\; I-beam.\; Here’s\; what\; each\; part\; of\; the\; designation\; means:\; •\hspace{1em}W:\; Indicates\; that\; the\; beam\; is\; a\; wide-flange\; (I-beam)\; section.\; •\hspace{1em}10:\; Refers\; to\; the\; nominal\; depth\; of\; the\; beam\; in\; inches\; (roughly\; 10\; inches\; deep).\; •\hspace{1em}54:\; Indicates\; that\; the\; beam\; weighs\; 54\; pounds\; per\; linear\; foot.\; So\; when\; you\; see\; a\; beam\; labeled\; W10X54,\; it\; means\; that\; every\; foot\; of\; that\; beam\; weighs\; 54\; pounds.\; This\; information\; is\; used\; by\; engineers\; to\; calculate\; loads,\; deflections,\; and\; other\; structural\; properties\; when\; designing\; a\; structure.''$

$\mathrm{Explain}\left(\mathrm{further}\right)$

$A designation like “W10×54” is read as “W-ten by fifty‐four” and refers to a standard American rolled I‐beam (also called a wide-flange beam). Here’s what the designation tells you and what “weight” means in this context: 1. W-shape: The “W” indicates that the beam is a wide-flange beam. These beams are commonly used in structural steel construction and are known for their efficient distribution of material along the flanges and web. 2. Nominal Depth: The number “10” roughly indicates the nominal depth of the beam in inches. In other words, the overall depth (from the top of the upper flange to the bottom of the lower flange) is approximately 10 inches. (Keep in mind that the actual measured depth might be slightly different from the nominal value.) 3. Weight per Unit Length: The “54” means that the beam weighs 54 pounds per foot of length. This is a standardized metric provided in steel manuals (like the AISC Steel Manual). When you see “W10×54,” you know that every one-foot segment of the beam has a weight of 54 pounds. This specific weight plays an important role in design considerations such as:   • Dead load calculations for buildings and bridges.   • Transportation and erection considerations.   • Overall structural analysis where self-weight is a factor. 4. Additional Properties: Although the weight per foot is one key characteristic, W-shapes are also defined by other properties such as:   • Moment of Inertia (about major and minor axes).   • Section Modulus.   • Radius of Gyration.   • Dimensions of the flanges and web. These properties are crucial for determining the beam’s bending, shear, and overall load-carrying capacity. 5. Practical Use: Knowing the weight per foot allows engineers to calculate the total weight of a beam for a given span (for example, a 20-foot length of W10×54 would have a dead weight of 20 ft × 54 lb/ft = 1080 lb). This helps in properly designing supports and connections, as well as in planning handling and installation logistics. In summary, the “Weight of a W10×54 beam” refers to its standard weight per foot (54 lb/ft), and this, along with its nominal 10-inch depth and other sectional properties, makes it a common and well-understood choice in structural steel design and construction.$

The OpenAI website.

The NaturalLanguage:-Explain command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.

The NaturalLanguage:-Explain command was updated in Maple 2025.