To multiply decimals, first multiply as if there is no decimal. Next, count the number of digits after the decimal in each factor. Finally, put the same number of digits behind the decimal in the product.

Arrange the numbers one on top of the other and line up the place values in columns. The number with the most digits is usually placed on top as the multiplicand. Starting with the ones digit of the bottom number, the multiplier, multiply it by the last digit in the top number. Write the answer below the equals line.

Definition of a Rational Number For example, 0.5 is a rational number. It is not a whole number, natural number, or integer, but it can be expressed as 1/2, which a fraction of two other integers: 1 is the numerator and 2 is the denominator. So, 0.5, or 1/2, is a rational number.

Answer in 60sInputResultKey moments in videoFrom 00:00 -- Intuitive IdeaFrom 01:54 -- DefinitionFrom 03:31 -- Graphing by HandFrom 06:10 -- Graphing by ComputerFrom 09:12 -- Vector Fields in 3DKey moments in videoFrom 01:25 -- Changing the number of vectors along the x and y axesFrom 01:46 -- Show the flow line from a particular pointFrom 02:21 -- Change the color of the vectorsRelated queriesWhich function will you use to plot a vector of a field?Use VectorDensityPlot and StreamDensityPlot to visualize the field densities: Copy to clipboard. Use VectorPlot3D to plot a three-dimensional vector field (vectors are colored depending on their…

Answer in 60sInput interpretationSeries expansion at x=0Related queriesWhat is the linear approximation of a function?The linear approximation of a function f(x) is the linear function L(x) that looks the most like f(x) at a particular point on the graph y = f(x).How do you calculate approximate value?If a number has digits to the right of the decimal less than 5. Then just drop the digits to the right of the decimals. The number so obtained will be the approximated value.What is the linearization equation?The Linearization of a function f(x,y) at (a,b) is L(x,y) = f(a,b)+(xâˆ’a)fx(a,b)+(yâˆ’b)fy(a,b). This is very similar to…

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Answer in 60sInput interpretationResultAlternate formsAlternate form assuming Ï‰>0Trigonometric transformRelated queriesWhat is the Fourier transform of sine wave?The Fourier Transform of the Sine and Cosine Functions Equation [2] states that the fourier transform of the cosine function of frequency A is an impulse at f=A and f=-A. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A.What is sine Fourier transform?In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used…

Answer in 60sInput interpretationRelated queriesWhat is Dirac delta function in quantum mechanics?The Dirac delta function is the name given to a mathematical structure that is intended to represent an idealized point object, such as a point mass or point charge. It has broad applications within quantum mechanics and the rest of quantum physics, as it is usually used within the quantum wavefunction.What is a delta function in calculus?The delta function is a generalized function that can be defined as the limit of a class of delta sequences. The delta function is sometimes called "Dirac's delta function" or the "impulse symbol"…

Answer in 60sInput interpretationInfinite sumSum convergencePartial sum formulaPartial sumsSeries representationsKey moments in videoFrom 06:39 -- Grand FinaleFrom 08:53 -- ConclusionKey moments in videoFrom 00:42 -- Analyzing the SeriesFrom 01:12 -- Comparison TestFrom 04:48 -- Why is this significant?From 06:38 -- Geometric Series ExampleFrom 07:27 -- What does this convergent series converge to?Related queriesHow do you find the sum of n terms?The formula to find the sum of n terms in AP is Sn = n/2 (2a+(nâˆ’1)d), in which a = first term, n = number of terms, and d = common difference between consecutive terms.Is the sequence 2 n convergent?So…

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Answer in 60sInputResultDiagonalizationKey moments in videoFrom 00:45 -- Create a matrixFrom 01:31 -- Convert matrix into transpose matrixFrom 03:19 -- Inverse matrix in mathematicaFrom 04:34 -- Summary of what we have seen so farRelated queriesWhat is a transpose operator?In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).What is the transpose of a tensor?To transpose a tensor, we need two dimensions to be transposed. If a tensor is…

Answer in 60secInput interpretationMath solution sin^2(x) graph Graph sin^2(x) graph Related questionsWhat is the period of the graph of the function y sin 2x?The equation is y=sin2x , so it is in the form y=sin(bâ‹…x) The period of this graph is 2Ï€b , which equals 2Ï€2 or Ï€ . Therefore, the "important points" (quarter points) when graphing this function will be Ï€4 apart.What is the amplitude of sin 2x?(c) y = sin 2x has amplitude 1, period Ï€ and phase shift 0.What is the formula for sin 2x?so that sin2x = 2 sin x cos x.What is sin squared of…

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Answer in 60sInputResultKey moments in videoFrom 00:07 -- Definition of a quadratic residueFrom 00:41 -- Example Checking if v is a Quadratic ResidueFrom 01:57 -- How to check your answer?Related queriesWhat are the quadratic residue of 7?Thus 1,2,4 are quadratic residues modulo 7 while 3,5,6 are quadratic nonresidues modulo 7.How do you calculate residue?We compute the residues at each pole: At z = i: f(z) = 1 2 Â· 1 z âˆ’ i + something analytic at i. Therefore the pole is simple and Res(f,i)=1/2. At z = âˆ’i: f(z) = 1 2 Â· 1 z + i + something…

Answer in 60sIndefinite integralRelated queriesWhat is an improper integral examples?For example, âˆ« 0 1 1 x d x \displaystyle\int_0^1 \dfrac{1}{\sqrt x}\,dx âˆ«01x 1dxintegral, start subscript, 0, end subscript, start superscript, 1, end superscript, start fraction, 1, divided by, square root of, x, end square root, end fraction, d, x is an improper integral.What is improper integral of first kind?An improper integral of the first kind is an. integral performed over an infinite domain, e.g. Z 1. a. f(x) dx.How do you find the improper integral?Let f(x) be continuous over (a,b]. Then, âˆ«baf(x)dx=limtâ†’a+âˆ«btf(x)dx. In each case, if the limit exists, then…

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Answer in 60sInput interpretationResultDimensionsMatrix plotMatrix rankNullityPseudoinverseRelated queriesWhat is row reduction?Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things.What is the use of reduce command in Mathematica?Reduce[expr,vars,dom] restricts all variables and parameters to belong to the domain dom. If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.How do you type matrix?To enter a matrix, use…

Answer in 60sInput interpretationResultKey moments in videoFrom 00:03 -- Solving Linear Systems with MathematicaFrom 00:11 -- Solve 3 EquationsFrom 01:54 -- Solve for a different variableFrom 02:16 -- (B)From 02:23 -- Variables, 3 equationsFrom 02:39 -- Using a Matrix (Not the Solve Function)From 03:29 -- Using RowReduce to Solve the SystemFrom 04:11 -- (Reduced form)From 04:59 -- Solving a system with only 1 solutionRelated queriesWhy is Gauss-Jordan method used?Gaussian Elimination and the Gauss-Jordan Method can be used to solve systems of complex linear equations. For a complex matrix, its rank, row space, inverse (if it exists) and determinant can all…

Answer in 60sInput interpretationGlobal maximumKey moments in videoFrom 00:53 -- First step: Entering the function in the calculatorFrom 03:43 -- How to draw a sketch of the graphFrom 09:56 -- Answering the question of absolute minimumFrom 11:37 -- Slope Number Line ExampleFrom 14:09 -- Summary & ConclusionRelated queriesWhat is absolute extremum?An absolute extremum (or global extremum) of a function in a given interval is the point at which a maximum or minimum value of the function is obtained. Frequently, the interval given is the function's domain, and the absolute extremum is the point corresponding to the maximum or minimum value…

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