8. Numbers and calculations

A. Decimals and fractions

A manufacturer is thinking about giving both metric measurements (for example, millimetres) and imperial measurements (for example, inches) in its product specifications. One of the company's engineers is giving his opinion on the idea in a meeting.

'One problem is, when you convert from metric to imperial you no longer have whole numbers - you get long decimal numbers. For example, one millimetre is nought point nought three nine three seven inches as a decimal. So to be manageable, decimals have to be rounded up or down. You'd probably round up that number to two decimal places, to give you zero point zero four. Now, you might say the difference is negligible - it's so small it's not going to affect anything. But even if it's just a tiny fraction of a unit - one hundredth of an inch (1/100), or one thousandth of an inch (1/1000) - and those numbers are then used in calculations, the rounding error can very quickly add up to give bigger inaccuracies.' 

Note: See Appendix III on page 100 for a list of metric and imperial units.

*deci·mal [ ˈdesɪml 데씨멀] 형용사 1.십진법의 the decimal system 십진법명사 1.소수 The decimal 0.61 stands for 61 hundredths. 소수 0.61은 100분의 61을 나타낸다. 

*frac·tion [ ˈfrækʃn 프랙션 ] 명사 1.부분, 일부 Only a small fraction of a bank’s total deposits will be withdrawn at any one time. 언제든 어느 한 번에는 은행 총 예금액의 적은 일부만을 인출할 수 있게 될 것이다. 2.분수
*met·ric [ ˈmetrɪk ]형용사 1.미터법의 metric units/measurements/sizes 미터법 단위/측정/치수 2. 미터법에 따라 만들어진[측정된] These screws are metric. 이 나사못들은 미터법에 따라 만들어졌다.
*im·per·ial [ ɪm'pɪriəl 임피리얼] 형용사 [명사 앞에만 씀]1. 제국의; 황제의 the imperial family / palace / army 황실/황궁/제국의 군대 2.(도량형이 영국의) 야드·파운드법에 의한
*whole number 명사 1.수학 정수, 범자연수
*nought [ nɔːt 나앗ㅌ] 명사 1.C, U 英 (zero 美, 英) 0, 영 A million is written with six noughts. 100만에는 0을 6개 쓴다. nought point one 0 대 1
*to be manageable 취급하기/처리하기 쉽게 하기위해
*man·age·able [ ˈmænɪdʒəbl ]형용사 1.관리[감당/처리]할 수 있는 The debt has been reduced to a more manageable level. 부채가 더 감당하기 쉬운 수준으로 줄었다. Cleanup after a hurricane is difficult, but manageable. 폭풍이 지나간 후 복구작업은 힘들긴 하지만 할 만하다. The work has been divided into smaller, more manageable sections. 그 작업은 좀 더 작고 다루기 쉬운 부분들로 나누어졌다.
*round sth (up/down) (to sth) 반올림[반내림]하다 (=round off) Round up the number to make it easier to calculate. 계산하기 쉽게 숫자를 반올림하여라. Round off the numbers to two decimal places. 소수 둘째 점에서 반올림 하시오 Round off to a whole number. 소수점 이하는 반올림하시오 Round off below the decimal point. 소수점 이하는 반올림해라
*neg·li·gible [ ˈneɡlɪdʒəbl 네그리져블 ] 형용사 1.(중요성·규모가 작아) 무시해도 될 정도의 The cost was negligible. 비용은 무시해도 될 정도였다. a negligible amount 무시해도 될 정도의 양
*af·fect [ əˈfekt ] 1.타동사 [VN 흔히 수동태로] 영향을 미치다 How will these changes affect us? 이들 변화가 우리에게 어떤 영향을 미칠까?
**affect 동사. ‘~에 영향을 주다[미치다]’: Does television affect children’s behaviour? 텔레비전이 아동들의 행동에 영향을 미치는가?
***effect 명사. ‘결과; 영향’: Does television have an effect on children’s behaviour? 텔레비전이 아동들의 행동에 영향을 미치는가?
*rounding error 1.(전산학) 반올림 오차 2.(수학) 끝수처리오차
*in·accur·acy [ ɪnˈækjərəsi ] 복수형 inaccuracies 부정확, 오류

 

B. Addition, subtraction, multiplication and division

During a TV programme about garden design, the presenter is explaining the calculations required to make a large setsquare which can be used for setting out.

To make one of these, you need to use Pythagoras's Theorem. So, a quick geometry lesson. Measure a length of timber for one of the sides adjacent to the right-angle. I've made this 3 feet long. Then square that number - 3 multiplied by 3 equals 9. Then do the same with the other side adjacent to the right-angle. I've made this one 4 feet long. Work out the square of that. So, 4 times 4 is 16. Then work out the sum of those two numbers - so if I add 16 to 9 ... 16 plus 9 is 25. Then, calculate the square root of that. The square root of 25 is 5. That means the longest side hypotenuse - needs to be 5 feet long. And it doesn't matter what length you make the two adjacent sides - if the square of the hypotenuse is equal to the square of each of the adjacent sides, added together, you'll have a perfect right-angle.

Now you can also start by making the hypotenuse, square the length of that, then make one of the other sides, square the length of that, and then subtract one from the other. For this example, that would be 25 minus 16. So, 25 less 16 is 9. And the square root of 9 is 3, which gives me the remaining side. Alternatively, you can make both the adjacent sides equal - make them the same length. So, take the square of the hypotenuse, which is 25, divide that by 2, which is 12.5, then work out the square root of 12.5, which ... requires a calculator! That's why it's easiest to use a 3-4-5 triangle, like this, which conveniently works with whole numbers. And that's also why I'm measuring in imperial, because 3 feet by 4 feet by 5 feet is a practical size to work with.

 

*add·ition [ əˈdɪʃn 어디션 ] 1.덧셈 2.추가된 것 (美) to build a new addition onto a house 주택에 증축을 하다 3.추가부가 Pasta’s basic ingredients are flour and water, sometimes with the addition of eggs or oil. 파스타의 기본 재료는 밀가루와 물이고, 가끔 달걀이나 오일이 추가된다.
*sub·trac·tion [ səbˈtrækʃn 썹트랙션 ] 뺄셈
*multi·pli·ca·tion[ ˌmʌltɪplɪˈkeɪʃn 멀티프리케이션] 명사 1.U 곱셈; 증식 Multiplication of cells leads to rapid growth of the organism. 세포 증식은 유기체의 급속한 증가로 이어진다.
*div·ision [ dɪˈvɪʒn 디비젼] 명사 1.U, sing. 분할; 분배 cell division 세포 분열 the division of labour between the sexes 성별 노동력 분배 a fair division of time and resources 시간과 자원의 공평한 분배 the division of the population into age groups 연령대별 인구 분할 2.U 나눗셈 3.C, U (특히 사회·단체 내의) 분열 There are deep divisions in the party over the war. 그 전쟁을 두고 당내 분열이 심하다. the work of healing the divisions within society 사회 내부의 분열 치유 작업 divisions between rich and poor 빈부 간의 분열 social/class divisions 사회적/계층간 분열
*set square 명사 1.삼각자 = a triangle(美)
*Py·thag·oras' the·orem [ paɪ'θæɡərəsɪs 'θiːərəm 파이쌔그로씨스씨어럼] 1.기하 피타고라스의 정리
*geom·etry[ dʒi'ɑːmətri 지아-메트리 ] 1.기하학 2.기하학적 구조
*ad·ja·cent [ əˈdʒeɪsnt 어제이슨ㅌ ] 형용사1.(지역·건물 등이)인접한, 가까운 The planes landed on adjacent runways. 그 비행기들은 가까운 활주로에 착륙했다. Our farm land was adjacent to the river. 우리 농지는 강에 인접해 있었다. 문형~ (to sth)
*square 2.타동사[주로 수동태로] 제곱하다 4 squared equals 16. 4의 제곱은 16이다.
**the square of; The square of 7 is 49. 7의 제곱은 49이다.
**the square root of; The square root of 64 (√64) is 8 (8x8=64). 64의 제곱근(√64)은 8이다.
*add [ æd ] 2.(수·양을) 합하다[더하다] [VN] Add 9 to the total. 합계에 9를 더하라. If you add all these amounts together you get a huge figure. 이들 액수를 모두 합하면 엄청난 수치가 나온다. [also V] 문형~ A to B | ~ A and B (together)
*multi·ply [ ˈmʌltɪplaɪ ] 1.곱하다 [VN] 2 multiplied by 4 is / equals / makes 8. 2 곱하기 4는 8이다(2 × 4 = 8). 2.(수·양을) 크게 증가[증대]시키다; 크게 증가[증대]하다 [VN] Cigarette smoking multiplies the risk of cancer. 흡연은 암의 위험을 크게 증대시킨다. 3.생물 증식[번식]하다; 증식[번식]시키다 [VN] It is possible to multiply these bacteria in the laboratory. 이들 박테리아를 실험실에서 증식시키는 것이 가능하다.
**times 전치사 1.비격식 …으로 곱한 Five times two is / equals ten. 5 곱하기 2는 10이다. 명사1.pl.…배(倍)가 되는 three times as long as sth …보다 3 배 긴 three times longer than sth …보다 3 배 긴
*work something⇄out 1. ~을 계산[산출]하다 to work out the answer 답을 계산해 내다 2.~을 해결하다[(답을) 알아내다] to work out a problem 문제를 해결하다 3.~을 계획해[생각해] 내다 I’ve worked out a new way of doing it. 내가 그것을 하는 새로운 방법을 생각해 냈다.
*the sum of; The sum of 7 and 12 is 19. 7과 12의 합은 19이다
*hypot·en·use [ haɪ'pɑːtənuːs 하이파러뉴-스 ] 명사1.기하 (직각 삼각형의) 빗변
*sub·tract [ səbˈtrækt 썹트랙ㅌ ]동사 1.(수·양을) 빼다 [VN] 6 subtracted from 9 is 3 9에서 6을 빼면 3이다. 문형 ~ sth (from sth)
**minus 전치사 1. …을 뺀[제외한] Seven minus three is four (7 − 3 = 4). 7 빼기 3은 4이다.
***less [ les ] 한정사, 대명사 1.(불가산 명사와 함께 쓰여) 더 적은[덜한]; 더 적은[덜한] 것 less butter / time / importance 더 적은 버터/시간/중요성 It is less of a problem than I’d expected. 그것은 내가 예상했던 것보다 문젯거리가 덜 된다. 부사 1.더 적게, 덜하게 less expensive / likely / intelligent 덜 비싼/그럴듯한/지적인 I read much less now than I used to. 나는 예전보다 독서를 훨씬 더 적게[덜] 한다. 전치사 1.…을 빼고[계산에 넣지 않고] a monthly salary of $2,000 less tax and insurance 세금과 보험료를 공제하지 않은 $2 000의 월급 25 less 16 is 9
*div·ide [ dɪˈvaɪd ] 동사 1.(여러 부분들로) 나뉘다[갈라지다]; 나누다[가르다] [V] The cells began to divide rapidly. 그 세포들은 빠르게 갈라지기[분열하기] 시작했다. [VN] A sentence can be divided up into meaningful segments. 하나의 문장은 여러 개의 의미 있는 조각들로 나뉠 수 있다. 문형~ (sth) (up) (into sth) 2. 타동사 [VN] (몫을) 나누다 Jack divided up the rest of the cash. 잭이 나머지 현금을 다 나누었다. We divided the work between us. 우리는 그 일을 우리끼리 나누었다. 4. 타동사 격식 [VN]A와 B를 갈라놓다 Can it ever be right to divide a mother from her child? 어미와 자식을 갈라놓는 일이 과연 옳을 수가 있는가? 7.타동사 [VN] (사람들을) 분열시키다 The issue has divided the government. 그 쟁점이 정부를 분열시켜 놓았다. 8.(~을) ~으로 나누다 [VN] 30 divided by 6 is 5. 30나누기 6은 5이다. [also V] 문형 ~ (sth) by sth

 

8.1 Write the numbers in words. Look at A opposite to help you.
1) 1.793                  one point seven nine three
2) 1/100 mm           one hundredth of a millimetre
3) 1/1000 mm         one thousandth of a millimetre
4) 0                         nought or zero

 

8.2 Complete the descriptions of the numbers using words from A opposite.
1) 0.25 = 1/4             The first number is a decimal, and the second is a fraction
2) 0.6368 ~~ 0.637   The second number is rounded up to three decimal places
3) 7.5278 ~~ 7.5       The second number is rounded down to one decimal place
4) 8, 26, 154             The numbers aren't fractions or decimals. They're whole numbers.   
5) Error: 0.00001%  The error is so small that it's negligible.
6) 0.586 kg x 9,000 = 5,274 kg / 0.59 kg x 9,000 = 5,310 kg   This difference is the result of a rounding error.

 

8.3 Complete the calculations using the words in the box. Sometimes there is more possible answer. Look at B opposite to help you.

1) 14+ 8 = 22 Fourteen plus eight equals twenty-two.
2) 100 × 20 = 2,000 One hundred times twenty is two thousand.
3) 7 x 11 = 77 Seven multiplied by eleven equals seventy-seven.
4) 400 ÷ 8 = 50 Four hundred divided by eight equals fifty.
5) 95+2=97 The sum of ninety-five and two is ninety-seven.
6) 8²=64 The quare of eight is sixty-four.
7) 50-30=20 if you substract thirty from fifty, it equals twenty.
8) √100 = 10 The square root of a hundred is ten.
9) 11² =  121 Eleven squared is a hundred and twenty-one.
10) 48-12=36 Forty-eight minus / less twelve equals thirty-six.

 

8.4 Use your knowledge of basic geometry to complete the sentences.

      Use one or two words from B opposite to fill each gap.

1) The sum of the three angles in a triangle equals 180 degrees. 
2) The area of a circle is equal to the square of its radius times / multiplied by 3.14.
3) The area of a right-angle triangle is equal to the length of one adjacent side, times / multiplied by the length of the other adjacent side, divided by two.

4) The length of each side of a square is equal to the square root of the square's area.
5) If each angle in a triangle is 60°, then the lengths of its sides are equal.