86 . 73 × 1 . 265 × 7607

Épreuve de setion européenne

Mantissae

"Mantissae"probably seemsas arhai to today's readersasa starterrank onthe frontof an au-

tomobile,but until1960 or soeveryhigh-shoolsiene student wastaughtthe loreof logarithms,and

in partiular how to use "ommon" (base-10) logarithmitables in alulation. Theiruse involved the

separationofalogarithminto twoparts:itsintegerpart(theharateristi)anditsfrationalpart(the

mantissa).Here isanexample:

Suppose, before the days of hand-held alulators, you needed a rapid way to multiply four-digit

numbers, and to divide that produt by another four-digit number, with an answer aurate to three

digits.Say

86 . 73 × 1 . 265 × 7607

. 3018 .

Proedure:Youthinkofeahofthenumbersasapowerof10timesanumberbetween1and10:

86 . 73 = 10 ¹ × 8 . 673

^;

1 . 265 = 10 ⁰ × 1 . 265

^;

7607 = 10 ³ × 7 . 607

^and

. 3108 = 10 ^{− 1} × 3 . 108 .

Whenyoutakelogarithms,sine

log( ab ) = log a + log b

,

log(86 . 73) = 1 + log(8 . 673)

^;

log(1 . 265) = 0 + log(1 . 265)

^;

log(7607) = 3 + log(7 . 607)

^and

log( . 3018) = − 1 + log(3 . 108) .

The seond term in eah of the logs is a number between0 and 1 : this will be the mantissa; the

leadingtermistheharateristi.Toobtainthelogoftheanswerthatwewant,

log(86 . 73) + log(1 . 265) + log(7607) − log( . 3018)

^,wemaketwoalulations.Firstweaddorsubtrattheharateristis;thisisan integer alulation. Inthis ase the total is 5. Then you onsult afour-plae logarithmitable for the

mantissae:

log(8 . 673) = . 93817

^;

log(1 . 265) = . 10209

^;

log(7 . 607) = . 88121

^and

log(3 . 018) = . 47972 .

Themantissaetotal(withsigns)to

1 . 44175

^.Youhopothe"1"andaddittotheharateristi.The log tablegives

log(2 . 765) = . 44170

^and

log(2 . 766) = . 44185

^{.Sineyouonlyexpet3plaes ofauray,}

youantake

2 . 765

^{asthemantissa}ontributionto theprodut, whihyoualulateas

10 ⁵⁺¹ × 2 . 765 = 2765000

^.

FromSimonNewomband"NaturalNumbers",http://www.ams.orgbyTonyPhillips

Questions

1. Whywerelogarithmitableseverneeded,andwhatinformationdidtheyshow?

2. Explainpreiselywhat theharateristisandthemantissaearein thismethod.

3. Explainwhytheresultfortheharateristisis5.

4. Usethepropertiesofthelogarithmandtheexampleto showthatthemethodisorret.

5. Use the method to get an approximate value of

log(7629 × 52 . 41)

^{. The values for the mantissae}

shallbegivenbythealulator,with 5DP.