Mathematica 9x By Again 11 [PORTABLE]

Generally accepted mathematical notation uses the capital letter I to denote identity matrices, matrices of various sizes with ones on the main diagonal and zeros elsewhere. These matrices have the property that AI = A and IA = A whenever the dimensions are compatible.

Therefore, when $q = 5$ for $\Delta q = 0.05$ and $0.01\text{,}$ we find that the difference quotient is equal to $-2.005$ and $-2.001\text{,}$ respectively. Therefore, when the quantity of tires demanded is between $5000$ and $5050\text{,}$ the unit price is decreasing by approximately $$2.00$ per $1000$ tires. And when the quantitity of tires demanded is between $5000$ and $5010\text{,}$ we again get that the unity price is decreasing by approximately $$2.00$ per $1000$ tires.

In the example above, the user first computes the value of 2.1^3. Next, the user would like to add the value of 1.1^2 too this result. Instead of typing 2.1^3 over again, the user is able to simply enter 1.1^2 and then use the % command. Mathematica interprets this as 1.1^2+2.1^3 and produces the desired result. While this example is fairly simple, using % becomes more and more useful when dealing with larger numbers or equations that would be tiresome to retype.

I have been troubled by this as well (this isn't an answer by the way). The most promising thing is to wrap the expression in a row of some width smaller than your document's. This works fine in terms of formatting within mathematica (with the definitions of @Nasser's answer):

so here you can do ToString@% and then play with StringReplace. This again is not satisfactory as it's more likely that if you have an expression with many terms you won't have something as simple as a sum and trying to make the above work in a general setting will probably take more time than you doing the editing in $\LaTeX.$

If you had a notebook with filename "c" then you could use this code to save a LaTeX document. By adjusting the window size in mathematica, you should see that the latex output wraps accordingly. In my case I was formatting Input cells which had very long commands, and they stretched off the page until I resized the window before export.

Successful teaching and learning of mathematics play an important role in ensuring that students have the right skills required to compete in a 21st century global economy. When properly implemented and coupled with opportunities for students to engage in mathematical investigation, communication and problem solving, rigorous mathematics standards hold the promise of elevating the mathematical knowledge and skill of every learner to levels competitive with the best in the world, of preparing our college entrants to undertake advanced work in the mathematical sciences, and of readying the next generation for the jobs their world will demand.

The math standards provide clarity and specificity rather than broad general statements. The standards draw on the most important international models for mathematical practice, as well as research. They endeavor to follow the design envisioned by William Schmidt and Richard Houang (2002), by not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles (coherence) such as place value and the laws of arithmetic to structure those ideas.

The Operations and Algebraic Thinking domain deals with the basic operations, the kinds of quantitative relationships they model, and consequently the kinds of problems they can be used to solve as well as their mathematical properties and relationships. Although most of the standards organized under this heading involve whole numbers, the domain includes concepts, properties, and representations that extend to other number systems, to measures, and to algebra.

Like core knowledge of number, core geometrical knowledge seems to be a universal capability of the human mind. Geometric and spatial thinking are important in and of themselves, because they connect mathematics with the physical world, and play an important role in modeling phenomena whose origins are not necessarily physical (i.e. networks or graphs). They are also important because they support the development of number and arithmetic concepts and skills. Thus, geometry is essential for all grade levels for many reasons: its mathematical content, its roles in physical sciences, engineering, and many other subjects, and its strong aesthetic connections.

An expression is a phrase in a sentence about a mathematical or real-world situation. As with a facial expression, you can read a lot from an algebraic expression without knowing the story behind it. It is a goal of this domain for students to see expressions as objects, and to read both the general appearance and fine details of algebraic expressions.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

When we reach 6174 the operation repeats itself, returning 6174 every time. We call the number 6174 a kernel of this operation. So 6174 is a kernel for Kaprekar's operation, but is this as special as 6174 gets? Well not only is 6174 the only kernel for the operation, it also has one more surprise up its sleeve. Let's try again starting with a different number, say 1789.

The table below shows the results: every four digit number where the digits aren't all equal reaches 6174 under Kaprekar's process, and in at most seven steps. If you do not reach 6174 after using Kaprekar's operation seven times, then you have made a mistake in your calculations and should try it again!

We have seen that all three digit numbers reach 495, and all four digit numbers reach 6174 under Kaprekar's operation. But I have not explained why all such numbers reach a unique kernel. Is this phenomenon incidental, or is there some deeper mathematical reason why this happens? Beautiful and mysterious as the result is, it might just be incidental.

This is a very beautiful puzzle and you might think that a big mathematical theory should be hidden behind it. But in fact it's beauty is only incidental, there are other very similar, but not so beautiful, examples. Such as:

even further you can take 9 down to 3. the mystery of 5 is in the sacred geometry of the golden ratio.which is a simple mathematical explanation of remaining numbers and the outward spiral and unfolding of the universe mathematically. five is a spiral symbolically. and in five there are three elements and two polar opposites thus 2:3=5 or 1:1:2:3:5 two and three make five but not one and one. this on whole two polars and three elements.?I'd like to know more......

Binomials are expressions with only two terms being added.2x^2 - 4x is an example of a binomial. (You can say that a negative 4x is being added to 2x2.)First, factor out the GCF, 2x. You're left with 2x (x - 2). This is as far as this binomial can go. Any binomial in the form 1x +/- n cannot be factored further.When you have a binomial that is a variable with an even exponent, added to a negative number that has a square root that is a natural number, it's called a perfect square.x^2 - 4 is an example of this. It can be expressed as the product of the square root of the variable plus the square root of the positive constant, and the square root of the variable minus the square root of the positive constant.Huh?Basically, take the square root of the variable. You'll end up with x. Then square root the 4. You'll end up with 2. If you add them together, you'll get x+2. Subtract them, and you'll get x-2. Multiply the two, and you'll get (x+4)(x-4). You've just factored a perfect square.If you multiply (x+2)(x-2) together using FOIL, you'll end back up with x^2-4.(FOIL: First Outer Inner Last, a way of multiplying two binomials together. Multiply the first terms of the binomials (x and x in this case), then the outer two (x and -2), then the inner two (2 and x), then the last terms (2 and -2), then add them all up. x^2 - 2x + 2x - 4 = x^2 - 4.)This can be done again if one of the binomials is a perfect square, as in this instance:x^4 - 16 = (x^2 + 4) (x^2 - 4) = (x^2 + 4) (x + 2) (x - 2).This can be factored further if you bring in irrational numbers, see step [9].How to factor binomials in the form of (x^3+ b^3):Just plug into (a - b) (a^2 +ab + b^2). For example, (x^3 + 8) = (x - 2) (x^2 + 2x + 4).How to factor binomials in the form of (x^3- b^3):Plug into (a + b) (a^2 - ab + b2). Note that the first two signs in the expression are switched.(x^3 - 8) = (x + 2) (x^2 - 2x + 4).Both examples can be factored further once you learn how to factor trinomials in step [4]. 2b1af7f3a8