By J. Pitha, G. Smets, D. Wohrle
With contributions through quite a few specialists
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20 Maths for Chemists enough by typing the numbers into a calculator or adding them in our head, to obtain the result 210, but the process becomes somewhat more tedious. Now, if we want to sum the sequence of integers from 1 to some, as yet unspecified, upper limit, denoted by the letter n, we need a formula that allows us to evaluate this sum without actually having to add each of the numbers individually. We can accomplish this as follows: Write down the sum of the first five integers, 1 to 5 , from highest to lowest, and introduce the symbol S5 to represent this sum: s 5 =5 0 +4 + 3 +2 + I Repeat the exercise by summing the same five integers from lowest to highest: 5’5 = 1 +2 + 3 +4 + 5 Add the two expressions to obtain: 2S5 = 6 + 6 + 6 + 6 + 6 = 5 x 6 = 30 +- S5 = 15 where the symbol +- means “implies”.
For example, we have 8 =n/2 rad or 8 = 90" or, equivalently, B/rad = 4 2 or O/" = 90. 15). The line joining the point on the circle to the origin is of length r, and equal to the radius of the circle. 15 The angle 8 represented in terms of a circle placed on a Cartesian coordinate system A zero value for the angle corresponds to the point lying on the positive x-axis. The angle increases in a positive sense as the point circulates in an anticlockwise direction; circulation in a clockwise sense is indicated by a negative value of 6.
For example, in 3x2 - 12xy 3 the numerator has 3 and x as common factors, whilst the denominator has 3. Since the denominator and numerator both have the common factor 3, this may be cancelled from each term to give: x2 -4xy 1 2 = x -4xy which simplifies further to x(x -4y), once the common factor x has been removed from each term. In this case, the rational expression reduces to a simple expression. e. ). 3 Polynomials A is represented by a sum of symbols raised to different powers, each with a different coefficient.
Unusual Properties Of New Polymers by J. Pitha, G. Smets, D. Wohrle