Mathematics and the physical world by Morris Kline

By Morris Kline

Stimulating account of improvement of uncomplicated arithmetic from mathematics, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. additionally describes how math is utilized in optics, astronomy, movement less than the legislations of gravitation, acoustics, electromagnetism, different phenomena.

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This is in fact the case and we can justify it as follows: Since then 4 x (-5) -4x(-5) = -20 =-(-20) = +20 ( - 4 ) * ( - 5 ) = 20 Hence By similar reasoning, we can show that the same results hold for division and so we may summarise as follows. When numbers are involved in multiplications or divisions: If the signs are the same, the result is positive. If the signs are different, the result is negative. 4 Evaluate the following: (a) (-^)x(-7) (b) +3X-11 (c) -7x5 (d) 9 +(-3) (e) ( - 7 ) + (-21) (0 ^ Solution (a) (-4) x (-7) = +(4 χ 7) = +28 = 28 (b) +3 x - 1 1 = - ( 3 x l l ) = - 3 3 Ch.

Is there an identity element for subtraction? Give a reason for your answer. Division: Rational N u m b e r s Division is the inverse of multiplication. g. 8-5-2 = 4 is the same as saying 8=4x2 or 8= 2x4 Division is obviously not commutative, since 2+8*8+2 8 + 2 is fine! It gives the answer 4 which is another natural number, but we are going again to have to extend the set of numbers since we have to find a meaning for 2 + 8. Think of placing extra small stones into the river (see below) between our main stepping stones.

11 + 12 χ 4 + 3 χ (5 + 3) 3 + (4 χ 5) * (3 + 4) χ (3 + 5) Subtraction Subtraction is the operation which performs the reverse or inverse process to addition. Thus choosing two natural numbers in a particular order we may write: 6 - 4 = 2 This is exactly equivalent to the statements 6 = 2+ 4 or 6 = 4 + 2 and we can think of the last two statements as being the result of having added 4 to both sides of the first equation. e. or (6-4)+ 4 = 2 + 4 = 6 4 +(6-4) = 4 + 2 = 6 Note that this might imply that subtraction is associative, but we must be very careful here, since we are mixing the binary operations ' + ' and Consider the following and remember that anything placed in brackets is done first: (7-4)- 2 = 3 - and 2=1 7 - (4 - 2) = 7 - 2 = 5 23 The Set of Real Numbers Ch.

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