[从头学数学] 第109节 有理数 小结与复习题

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了筑基初期的修炼，
这次要修炼的目标是[有理数 小结与复习题]。

正剧开始：

星历2016年02月25日 14:48:26, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起进行有理数的复习。

<span style="font-size:18px;">//题1
function myDraw() { 
	
    var config = new PlotConfiguration();    
    config.init();    
    config.setPreference();   
	config.setSector(1,1,1,1);
	config.axis1D(0, 0, 280);

	var r = 2*(280-10);
	//刻度 scale; graduation ; 
	var scale = r / 10;
	
	var array = [3.5,-3.5,0,2,-2,-1.6,-0.33,0.5];
	var size = array.length;
	
	array.sort(function(a, b) { return a-b;});
	plot.fillText(array.join('  <  '), -240, 60, 400);
	//存放点阵列
	var points = [];
	
	for (var i = 0; i < size; i++) {		
		points.push([array[i]*scale, 0]);
	}
	
	//由于shape采用的是弹出数据的方式，所以拷贝一个副本来进行操作
	var tmp = points;
	
	shape.pointDraw(tmp, 'green');
	
	for (var i = 0; i < size; i++) {
		plot.fillText(array[i].toString(), array[i]*scale-10, 20, 50);
	}

}
</span>

<span style="font-size:18px;">//题2
function myDraw() { 
	
    var config = new PlotConfiguration();    
    config.init();    
    config.setPreference();   
	config.setSector(1,1,1,1);
	config.axis1D(0, 0, 280);

	var r = 2*(280-10);
	//刻度 scale; graduation ; 
	var scale = r / 10;
	
	var array = [];
	
	for (var i = -2; i < 4; i++) {
		array.push(i);
	}
	
	var size = array.length;
	
	array.sort(function(a, b) { return a-b;});
	//plot.fillText(array.join('  <  '), -240, 60, 400);
	//存放点阵列
	var points = [];
	
	for (var i = 0; i < size; i++) {		
		points.push([array[i]*scale, 0]);
	}
	
	//由于shape采用的是弹出数据的方式，所以拷贝一个副本来进行操作
	var tmp = points;
	
	shape.pointDraw(tmp, 'green');
	
	for (var i = 0; i < size; i++) {
		plot.fillText(array[i].toString(), array[i]*scale-5, 20, 50);
	}

}</span>

<span style="font-size:18px;">>>> 
-2, 绝对值是2，相反数是2，倒数是-0.5。
-0.67, 绝对值是0.67，相反数是0.67，倒数是-1.5。
5.5, 绝对值是5.5，相反数是-5.5，倒数是0.18。

#题3
def tmp(num):
    print('{0}, 绝对值是{1}，相反数是{2}，倒数是{3}。'.format(\
        round(num,2), round(abs(num),2), round(-num, 2), round(1/num, 2)));
    return;

if __name__ == '__main__':
    a = -2;
    b=-2/3;
    c = 5.5;
    tmp(a);
    tmp(b);
    tmp(c);</span>

<span style="font-size:18px;">#题5
-150+250 = 100, 
-15+(-23) = -38, 
-5-65 = -70, 
-26-(-15) = -11, 
-6*(-16) = 96, 
-1/3*27 = -9 = -900%, 
8/(-16) = -1/2 = -0.5 = -50%, 
-25/(-2/3) = 75/2 = 37.5, 
(-0.02)*(-20)*(-5)*4.5 = -9 = -900%, 
(-6.5)*(-2)/(-1/3)/(-5) = 39/5 = 7.8 = 780%, 
6+(-1/5)-2-(-1.5) = 53/10 = 5.3 = 530%, 
-66*4-(-2.5)/(-0.1) = -289, 
(-2)**2*5-(-2)**3/4 = 22, 
-(3-5)+3**2*(1-3) = -16, </span>

<span style="font-size:18px;">>>> 
245.635精确到小数点后1位是245.6。
175.65精确到小数点后0位是176。
12.004精确到小数点后2位是12.00。
6.5378精确到小数点后2位是6.54。

#近似数
def approximate(num, accuracy):  
    if accuracy > 0:  
        result = str(round(num, accuracy));  
    else:  
        result = str(round(num));  
    #对于要求保留小数的情况，由于一般末尾的0会被舍弃，  
    #导致精度位数不够，所以在此进行判断补充。  
    if accuracy > 0:  
        pos = result.find('.');  
        diff = accuracy-(len(result)-pos-1);  
        result += '0'*diff;  
    print('{0}精确到小数点后{1}位是{2}。'.format(num, accuracy, result));  

#题6
def tmp():
    approximate(245.635,1);
    approximate(175.65, 0);
    approximate(12.004,2);
    approximate(6.5378, 2);</span>

<span style="font-size:18px;">>>> 
100000000 用科学计数法表示是 1e+08
-4500000 用科学计数法表示是 -4.50e+06
692400000000 用科学计数法表示是 6.924e+11

#题7
def tmp():
    a = 100000000;
    print('{0} 用科学计数法表示是 {1:.0e}'.format(a, a));
    a = -4500000;
    print('{0} 用科学计数法表示是 {1:.2e}'.format(a, a));
    a = 692400000000;
    print('{0} 用科学计数法表示是 {1:.3e}'.format(a, a));
	
>>> 
-5
1</span>

<span style="font-size:18px;">#题8
def tmp():
    print(-2-abs(-3));
    print(abs(-2-(-3)));</span>

<span style="font-size:18px;">#题9
82+83+78+66+95+75+56+93+82+81 = 791, 
(82+83+78+66+95+75+56+93+82+81)/10 = 791/10 = 79.1, 

#题11
458-(-27.8-70.3+200+138.1-8+188) = 38, </span>

<span style="font-size:18px;">//题12
function myDraw() { 
	
    var config = new PlotConfiguration();    
    config.init();    
    config.setPreference();   
	config.setSector(1,1,1,1);
	config.axis2D(0, 0, 180);

	var array = [];
	array.push([15, -15*2]);
	array.push([60, -60*2]);
	array.push([115, -5*2]);
	
	var tmp = [].concat(array);
	
	shape.pointDraw(tmp, 'red');
	
	tmp = [].concat(array);
	shape.strokeDraw(tmp, 'red');
	

}</span>

<span style="font-size:18px;">>>> 
149600000.0 用科学计数法表示是 1.496e+08

#题13
def tmp():
    a = 1.4960*10**8;
    print('{0} 用科学计数法表示是 {1:.3e}'.format(a, a));</span>

<span style="font-size:18px;">>>> 
1 * 1 =  1
11 * 11 =  121
111 * 111 =  12321
1111 * 1111 =  1234321
11111 * 11111 =  123454321
111111 * 111111 =  12345654321
1111111 * 1111111 =  1234567654321
11111111 * 11111111 =  123456787654321
111111111 * 111111111 =  12345678987654321
1111111111 * 1111111111 =  1234567900987654321
11111111111 * 11111111111 =  123456790120987654321
111111111111 * 111111111111 =  12345679012320987654321
1111111111111 * 1111111111111 =  1234567901234320987654321
11111111111111 * 11111111111111 =  123456790123454320987654321
111111111111111 * 111111111111111 =  12345679012345654320987654321
1111111111111111 * 1111111111111111 =  1234567901234567654320987654321

#题16
def tmp():
    a = [];
    a.append(1);
    for i in range(15):
        b = a[-1];
        a.append(b*10+1);

    size = len(a);
    for i in range(size):        
        print('{0} * {0} =  {1}'.format(a[i], a[i]*a[i]));
		
	</span>

计算工具：

<span style="font-size:18px;">###
# @usage   百分数，分数， 小数各结果的运算
# @author  mw
# @date    2016年02月19日  星期五  10:42:16 
# @param
# @return
#
###
def percentCalc():
    
    # 可以将任意的四则运算表达式用分数进行计算，
    # 而不是将浮点结果简单转化成分数。
    fin = open('input.txt');
    fout= open('output.txt', 'a');

    #结果为整数，此时小数表示和分数结果相同。
    resultIsInt = False;
    #结果绝对值大于10， 此时用百分数表示不合适。
    noPercentExpr = False;
    
    for line in fin.readlines():
        if line[-1] == '\n':
            line = line[:-1];     
            
        if line == '':
            continue;
        elif line.startswith('#'):
            print(line);
            fout.write(line+'\n');
        else:
            try:
                lines = line.split(sep=' ');
                for i in range(len(lines)):
                    if lines[i]=='':
                        continue;

                    #消除空格
                    lines[i] = lines[i].replace(' ', '');
                    expCount = len(lines[i]);
                    expression = '';
                    
                    resultIsInt = False;
                    noPercentExpr = False;
                    originExpr = lines[i];

                    #看表达式中是否有百分号
                    havePercentOp = False;
                    if lines[i].find('%') != -1:
                        havePercentOp = True;

                    OpsArray = [];
                    if havePercentOp == True:
                        for j in range(expCount):
                            #操作符直接添加
                            if  isOps(lines[i][j]):
                                OpsArray.append(lines[i][j]);
                            #处理百分号
                            if lines[i][j] == '%':
                                if len(OpsArray) > 0 and OpsArray[-1] == '/':
                                    expression+='*100';
                                else:
                                    expression+='/100';
                            else:
                                expression+=lines[i][j];

                        lines[i] = expression;
                        expCount = len(lines[i]);
                        expression = '';                   

                    #小数结果
                    floatResult = 0;
                    #百分数结果
                    percentResult = 0;
                    
                    result = eval(lines[i]);
                    if (abs(result)>10):
                        noPercentExpr = True;
                        
                    if (abs(result - int(result)) < 0.000001):
                        resultIsInt = True;
                        #小数结果
                        floatResult = int(result);
                        percentResult = int(result)*100;
                    else:
                        #小数结果
                        floatResult = round(result, 3);

                        result2 = result * 100;
                        if (abs(result2 - int(result2)) < 0.000001):
                            percentResult = int(result2);
                        else:
                            percentResult = round(result2, 3);                 

                    #分割操作数
                    operands = [];
                    stmp = '';
                    indexBeg = 0;
                    indexEnd = 0;
                    for j in range(expCount):
                        if isOps(lines[i][j]):
                            if stmp != '':
                                operands.append(stmp);
                                stmp = '';
                        else:
                            stmp+=lines[i][j];
                    if stmp != '':
                        operands.append(stmp);
                    
                    print(operands);

                    #操作数修饰
                    operandCount = len(operands);
                    operandString = [];
                    #数字如1/7要转化成Fraction('1/7')这种样式才可以算出正确结果，引号不可以少掉。
                    for j in range(operandCount):
                        stmp = '';
                        stmp = 'Fraction(\''+operands[j]+'\')';
                        operandString.append(stmp);

                    #组装新表达式       
                    typechange = 1;
                    index = 0;
                    expression = '';      
                    
                    
                    for j in range(expCount):
                        #操作符直接添加
                        if  isOps(lines[i][j]):
                            #记录操作符
                            OpsArray.append(lines[i][j]);
                            expression+=lines[i][j];
                        else:
                            if j > 0 and typechange == 0 and isOps(lines[i][j-1]):
                                typechange = 1;
                            
                            #从操作数序列中选择对应操作数。
                            if typechange == 1:
                                if index > len(operandString):
                                    break;
                                expression += operandString[index];
                                index+=1;
                                typechange = 0;                 


                    #设置打印格式
                    if noPercentExpr == True:
                        if resultIsInt == True:
                            s = '{0} = {1}'.format(originExpr,eval(expression));
                        else:
                            s = '{0} = {1} = {2}'.format(originExpr,eval(expression), floatResult);
                    else:
                        if resultIsInt == True:
                            s = '{0} = {1} = {2}%'.format(originExpr,eval(expression),percentResult);
                        else:
                            s = '{0} = {1} = {2} = {3}%'.format(originExpr,eval(expression), floatResult, percentResult);

                    print(s, end=', ');
                    fout.write(s + ', ');
                print('\n');
                fout.write('\n');
            except:
                #traceback.print_exc();
                ex = '{0} 表达式有误，无法进行计算。'.format(lines[i]);
                print(ex);
                fout.write(ex);

    fout.write('\n');
    fout.close();
    fin.close();
</span>

本节到此结束，欲知后事如何，请看下回分解。