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Having-given-log-2-0-30103-find-the-position-of-the-first-significant-figure-in-2-37-




Question Number 58060 by Kunal12588 last updated on 17/Apr/19
Having given log 2 = 0.30103, find the position  of the first significant figure in 2^(−37) .
$${Having}\:{given}\:\mathrm{log}\:\mathrm{2}\:=\:\mathrm{0}.\mathrm{30103},\:{find}\:{the}\:{position} \\ $$$${of}\:{the}\:{first}\:{significant}\:{figure}\:{in}\:\mathrm{2}^{−\mathrm{37}} . \\ $$
Commented by Rasheed.Sindhi last updated on 17/Apr/19
−11.13811≠11^(−) .13811  −11.13811 is whole negative  i-e it is equal to  −11−.13811  While in 11^(−) .13811 only integer part is  negative. I-e 11^(−) .13811=−11+.13811  As −11.13811 is whole negative,  therefore in it .13811 is not mantissa  (and −11 is also not characteristics.)  because mantissa is always positive  To obtain mantissa and characteristics  from −11.13811 thefollowing process  is mentioned:       −11.13811=−11.13811+12−12  =(−11.13811+12)−12=.86189−12  =12^(−) .86189
$$−\mathrm{11}.\mathrm{13811}\neq\overline {\mathrm{11}}.\mathrm{13811} \\ $$$$−\mathrm{11}.\mathrm{13811}\:{is}\:{whole}\:{negative} \\ $$$${i}-{e}\:{it}\:{is}\:{equal}\:{to}\:\:−\mathrm{11}−.\mathrm{13811} \\ $$$${While}\:{in}\:\overline {\mathrm{11}}.\mathrm{13811}\:{only}\:{integer}\:{part}\:{is} \\ $$$${negative}.\:{I}-{e}\:\overline {\mathrm{11}}.\mathrm{13811}=−\mathrm{11}+.\mathrm{13811} \\ $$$${As}\:−\mathrm{11}.\mathrm{13811}\:{is}\:{whole}\:{negative}, \\ $$$${therefore}\:{in}\:{it}\:.\mathrm{13811}\:{is}\:{not}\:{mantissa} \\ $$$$\left({and}\:−\mathrm{11}\:{is}\:{also}\:{not}\:{characteristics}.\right) \\ $$$${because}\:{mantissa}\:{is}\:{always}\:{positive} \\ $$$${To}\:{obtain}\:{mantissa}\:{and}\:{characteristics} \\ $$$${from}\:−\mathrm{11}.\mathrm{13811}\:{thefollowing}\:{process} \\ $$$${is}\:{mentioned}: \\ $$$$\:\:\:\:\:−\mathrm{11}.\mathrm{13811}=−\mathrm{11}.\mathrm{13811}+\mathrm{12}−\mathrm{12} \\ $$$$=\left(−\mathrm{11}.\mathrm{13811}+\mathrm{12}\right)−\mathrm{12}=.\mathrm{86189}−\mathrm{12} \\ $$$$=\overline {\mathrm{12}}.\mathrm{86189} \\ $$
Commented by Kunal12588 last updated on 17/Apr/19
ohh... I got it sir thanks a lot.
$${ohh}…\:{I}\:{got}\:{it}\:{sir}\:{thanks}\:{a}\:{lot}. \\ $$
Commented by Kunal12588 last updated on 17/Apr/19
Answer given in the book :−  log  2^(−37) =−37 log 2 = −37 × 0.30103  =−11.13811=12^(−) .86189  Hence in 2^(−37)  there are 11 cyphers following  the decimal point, i.e. the first significant  figure is in the twelfth place of decimals.  My Question is why  and how they have done  the last line. −11.13811=11^(−) .13811=12^(−) .86189
$${Answer}\:{given}\:{in}\:{the}\:{book}\::− \\ $$$$\mathrm{log}\:\:\mathrm{2}^{−\mathrm{37}} =−\mathrm{37}\:\mathrm{log}\:\mathrm{2}\:=\:−\mathrm{37}\:×\:\mathrm{0}.\mathrm{30103} \\ $$$$=−\mathrm{11}.\mathrm{13811}=\overline {\mathrm{12}}.\mathrm{86189} \\ $$$${Hence}\:{in}\:\mathrm{2}^{−\mathrm{37}} \:{there}\:{are}\:\mathrm{11}\:{cyphers}\:{following} \\ $$$${the}\:{decimal}\:{point},\:{i}.{e}.\:{the}\:{first}\:{significant} \\ $$$${figure}\:{is}\:{in}\:{the}\:{twelfth}\:{place}\:{of}\:{decimals}. \\ $$$${My}\:{Question}\:{is}\:{why}\:\:{and}\:{how}\:{they}\:{have}\:{done} \\ $$$${the}\:{last}\:{line}.\:−\mathrm{11}.\mathrm{13811}=\overline {\mathrm{11}}.\mathrm{13811}=\overline {\mathrm{12}}.\mathrm{86189} \\ $$

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