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Tag: Real Numbers
Visualising Solid Shapes | Study
Real Numbers | Study
Visualising Solid Shapes | Study
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English Version Visualising SolidShapes | Speed Notes
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The circle, thesquare, the rectangle, the quadrilateral and the triangle are examples of plane figures; the cube, the cuboid, the sphere, the cylinder, the cone and the pyramid areexamples of solid shapes. (Scroll down to continue …)
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Plane figures areof two-dimensions (2-D) and the solid shapes are of three- dimensions (3-D). The corners of a solid shape are called its vertices; theline segments ofits skeleton areits edges; and itsflat surfaces areits faces. A net is a skeleton-outline of a solid that can be folded to make it. The same solid can haveseveral types ofnets. Solid shapes can be drawn on a flat surface (like paper) realistically. We call this 2-D representation of a 3-Dsolid. Two types ofsketches of asolid are possible: (a) An oblique sketch does nothave proportional lengths. Still it conveys all important aspects of the appearance of the solid. (b) An isometric sketch is drawn on an isometric dot paper, a sample of which isgiven at theend of thisbook. In an isometric sketch of the solidthe measurements kept proportional. Visualising solidshapesis a veryuseful skill. Youshould be ableto see ‘hidden’ parts of thesolid shape. Different sections of a solid can be viewed in many ways: (a) One way is to viewby cutting or slicing the shape, whichwould result in the cross- section of thesolid. (b) Another way isby observing a 2-D shadow of a 3-Dshape. (c) A third wayis to lookat the shapefrom different angles; the front-view, theside- view and thetop view canprovide a lotof information aboutthe shape observed.
19. When a grouping symbol preceded by ‘ sign is removed or inserted, thenthe sign of eachterm of thecorresponding expression ischanged (from ‘ + ‘ to ‘−’ and from‘− ‘ to + ‘).
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Real Numbers | Study
Pre-Requisires
Test & Enrich
English Version Real Numbers | Speed Notes
Notes For Quick Recap
Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm :
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder. (Scroll down to continue …).
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Introduction:
Euclid’s Division Lemma/Euclid’s Division Algorithm:
Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.
This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder.
NOTE:
1. Lemma is a proven statement used for proving another statement.
2. Euclid’s Division Algorithm can be extended for all integers, except zero i.e., b 0.
HCF of two positive integers :
HCF of two positive integers a and b is the largest integer (say d ) that divides both a and b(a>b) and is obtained by the following method :
Step 1. Obtain two integers r and q, such that a=bq+r, 0r<b.
Step 2. If r=0, then b is the required HCF.
Step 3. If r0, then again obtain two integers using Euclid’s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.
The Fundamental Theorem of Arithmetic :
Every composite number can be factorised as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.
Irrational Number :
A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).
OR
A number which cannot be expressed in the form of pq , q 0 and p, qI, will be an irrational number. The set of irrational numbers is generally denoted by Q.
NOTE:
1. The rational number pq will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2n5m, where n, m are some non-negative integers.
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