[从头学数学] 第139节 二次根式 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中期的修炼，
这次要修炼的目标是[二次根式 小结与复习题]。

正剧开始：

星历2016年03月18日 12:19:31, 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]一起研究[二次根式 小结与复习题]。

<span style="font-size:18px;">			'x>=-3',
			'x>1/2',
			'x<2/3',
			'x!=1',
			'10*5^[0.5]',
			'2*3x^[0.5]',
			'2*(2/3)^[0.5]',
			'(2/3)^[0.5]/a',
			'xy*2y^[0.5]',
			'(5a/6)^[0.5]a^[2]',
			</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	//config.setSector(1,1,1,1);  
	//config.graphPaper2D(0, 0, r);
	//config.axis3D(0, 0,0,180);  
	

	plot.setFillStyle('blue');
	var mathText = new MathText();
	var s = [
			'(24^[0.5]-(1/2)^[0.5])-((1/8)^[0.5]+6^[0.5])',
			'2*12^[0.5]*3^[0.5]/4/(5*2^[0.5])',
			'(2*3^[0.5]+6^[0.5])*(2*3^[0.5]-6^[0.5])',
			'(2*48^[0.5]-3*27^[0.5])/6^[0.5]',
			'(2*2^[0.5]+3*3^[0.5])^[2]',
			'(3/2*(5/3)^[0.5]-(5/4)^[0.5])^[2]',
		
		];
		var x =40, y=40;
		
		var len = s.length;
		for (var i = 0; i < len; i++) {
		
			if (s[i] == '') {
				if (x < 100) {
					x += 300;
					y-=30*3;
				}
				else {
					x = 20;
					y += 30;
				}
			}
			else {			
				mathText.print(s[i], x, y);
				y+=30;
			}
		}		
	
}</span>

<span style="font-size:18px;">def tmp():
    s = ['(24^[0.5]-(1/2)^[0.5])-((1/8)^[0.5]+6^[0.5])',
			'2*12^[0.5]*3^[0.5]/4/(5*2^[0.5])',
			'(2*3^[0.5]+6^[0.5])*(2*3^[0.5]-6^[0.5])',
			'(2*48^[0.5]-3*27^[0.5])/6^[0.5]',
			'(2*2^[0.5]+3*3^[0.5])^[2]',
			'(3/2*(5/3)^[0.5]-(5/4)^[0.5])^[2]'];
    for i in range(len(s)):
        s1 = s[i].replace('^[', '**');
        s1 = s1.replace(']', '');
        print('{0} = {1}'.format(s1, round(eval(s1), 3)));</span>

<span style="font-size:18px;">>>> 
(24**0.5-(1/2)**0.5)-((1/8)**0.5+6**0.5) = 1.389
2*12**0.5*3**0.5/4/(5*2**0.5) = 0.424
(2*3**0.5+6**0.5)*(2*3**0.5-6**0.5) = 6.0
(2*48**0.5-3*27**0.5)/6**0.5 = -0.707
(2*2**0.5+3*3**0.5)**2 = 64.394
(3/2*(5/3)**0.5-(5/4)**0.5)**2 = 0.67</span>

<span style="font-size:18px;">>>> f = lambda x : x**2+5*x-6;
>>> f(5**0.5-1);
1.7082039324993694
>>> f2 = lambda x : (7+4*3**0.5)*x**2+(2+3**0.5)*x+3**0.5;
>>> f2(2-3**0.5);
3.732050807568878
>>> I = lambda Q, R, t: (Q/R/t)**0.5;
>>> I(30, 5, 1);
2.449489742783178</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	//config.setSector(1,1,1,1);  
	//config.graphPaper2D(0, 0, r);
	//config.axis3D(0, 0,0,180);  
	

	plot.setFillStyle('red');
	var mathText = new MathText();
	var s = [
			'PI*R_[1]^[2] = PI*R_[0]^[2]/4 => R_[1] = 1/2*R_[0]', 
			'PI*(R_[2]^[2]-R_[1]^[2]) = PI*R_[0]^[2]/4 => R_[2] = 2^[0.5]/2*R_[0]',
			'R_[3] = 3^[0.5]/2*R_[0]',
		
			'OA = R_[0]',
			'OD = R_[1]',
			'OC = R_[2]', 
			'OB = R_[3]',
			
			
		];
		var x =40, y=40;
		var r1 = 40;
		
		var len = s.length;
		for (var i = 0; i < len; i++) {
		
			if (s[i] == '') {
				if (x < 100) {
					x += 300;
					y-=r1*3;
				}
				else {
					x = 20;
					y += r1;
				}
			}
			else {			
				mathText.print(s[i], x, y);
				y+=r1;
			}
		}		
	
}</span>

<span style="font-size:18px;">def tmp():
    s = ['(24^[0.5]-(1/2)^[0.5])-((1/8)^[0.5]+6^[0.5])',
			'2*12^[0.5]*3^[0.5]/4/(5*2^[0.5])',
			'(2*3^[0.5]+6^[0.5])*(2*3^[0.5]-6^[0.5])',
			'(2*48^[0.5]-3*27^[0.5])/6^[0.5]',
			'(2*2^[0.5]+3*3^[0.5])^[2]',
			'(3/2*(5/3)^[0.5]-(5/4)^[0.5])^[2]'];
    for i in range(len(s)):
        s1 = s[i].replace('^[', '**');
        s1 = s1.replace(']', '');
        print('{0} = {1}'.format(s1, round(eval(s1), 3)));
		
def tmp():
    for i in range(1, 189):
        n = (189*i)**0.5;
        if (abs(n-round(n)) < 0.0001):
            print(i);
            break;
			
>>> 
21</span>

为了增加这个下标，小伟的工具又进化了一下：

<span style="font-size:18px;">/**
* @usage   数学表达式，代数式的书写
* @author  mw
* @date    2016年03月12日  星期六  11:05:12 
* @param
* @return
*
*/
function MathText() {
	//上标标记形式为...^[内容]...
	//分数不进行处理， 根式不进行处理，都转成指数式进行
	//特殊数学符号设想加\[alpha]进行转义，待续
	//可以进行指数上标代数式的书写
	//可扩展下标，待续
	
	
	this.setNormalFont = function() {
		plot.setFont("normal normal normal 24px Times Lt Std");	
	}
	
	this.setScriptFont = function() {
		plot.setFont("italic normal bold 16px Dark Courier ");
	}
	
	this.print = function(text, xPos, yPos) {
		xPos = xPos ? xPos : 0;
		yPos = yPos ? yPos : 0;
		
		plot.save();
		
		
		var s = text ? text : '';
		
		if (s != '') {
			s = s.replace(/\/\//ig, '÷');
			s = s.replace(/>=/ig, '≥');
			s = s.replace(/<=/ig, '≤');
			s = s.replace(/!=/ig, '≠');
			s = s.replace(/pi/ig, 'π');
		}
		
		//字符串长度
		var len = s.length;
		//不同字体大小设置在此
		var r1 = 20;
		//单个字符暂存处
		var c;
		//文本显示位置
		var x = xPos, y = yPos;
		//正常文本暂存
		var s0 = '';
		//字符串打印长度
		var measure; 
		//记录上一个x位置,可记录三层
		var xMem = [x, x, x];
		//记录每一层的左括号位置
		var bracketPos = [x, x, x];
		//记录括号层次
		var bracketsLevel = 0;
		//记录根号层次
		var radicalLevel = 0;
		
		//设置正常字体
		this.setNormalFont();				
		
		for (var i = 0; i < len; i++) {
			if (s[i] == '_') {
				//下标开始
				//下标标记形式为..._[内容]...
				
				if (s0 != '') { //先把正常字符打印出
					if (r1 != 20) { //字体字号大小还在上标状态
						r1 = 20;
						this.setNormalFont();					
					}
					
					measure = plot.measureText(s0);
					plot.fillText(s0, x, y, measure);
					s0 = '';

					x += measure;
				
				}
				
				var subScript = '';
				var j = 0;
				for (j = i+1; s[j]!=']'; j++) {
					if (s[j] != '[') {
						subScript+=s[j];
					}
				}
				
				if (r1 != 10) {//正常字体状态，需要改为上标字体
					r1 = 10;
					this.setScriptFont();
					
				}
			
				measure = plot.measureText(subScript);
				plot.fillText(subScript, x, y+8, measure);
					
				if (j < len-1 && s[j+1] == '^') {
				
				}
				else {
					x += 1.2*measure;
				}
				
				i = j;
			
			}
			else if (s[i] == '^') {
				//上标开始
				//上标标记形式为...^[内容]...
				
				if (s0 != '') { //先把正常字符打印出
					if (r1 != 20) { //字体字号大小还在上标状态
						r1 = 20;
						this.setNormalFont();					
					}
					
					measure = plot.measureText(s0);
					plot.fillText(s0, x, y, measure);
					s0 = '';

					x += measure;
				
				}
					
				var upperScript = '';
				var j = 0;
				for (j = i+1; s[j]!=']'; j++) {
					if (s[j] != '[') {
						upperScript+=s[j];
					}
				}
				
				
				var x1, y1;
				if (i > 0 && s[i-1] == ')') {
					x1 = bracketPos[bracketsLevel], y1 = y-20-5*radicalLevel;
				}
				else {
					x1 = xMem[bracketsLevel], y1 = y-20-5*radicalLevel;
				}
					
				
				//二次根式
				if (upperScript == '1/2' || upperScript == '0.5') {					
					
					plot.save()
						.setLineWidth(1);
					plot.beginPath()
						.moveTo(x1-5, y+5)
						.lineTo(x1-8, y-3)
						.moveTo(x1-5, y+5)
						.lineTo(x1+5, y1)
						.moveTo(x1+5, y1)
						.lineTo(x, y1)
						.closePath()
						.stroke();
					plot.restore();
					
				}
				else {

					if (r1 != 10) {//正常字体状态，需要改为上标字体
						r1 = 10;
						this.setScriptFont();
						
					}
				
					measure = plot.measureText(upperScript);
					plot.fillText(upperScript, x, y-8, measure);
					
					if (j < len-1 && s[j+1] == '_') {
					
					}
					else {
						x += 1.2*measure;
					}
				}
				
				//直接跳跃过上标字符区段
				i = j;
			}
			else {
				c = s[i];
				
				if (c == ')') {
					s0 += c;
					bracketsLevel -= 1;
					
				}
				else if (c == '(') {
					//如果整个括号被开根式，根号在括号左边
					bracketPos[bracketsLevel] = x + plot.measureText(s0);					
					s0 += c;
					
					bracketsLevel+=1;
					//过了括号就是过了一道关，要刷新坐标
					xMem[bracketsLevel] = x + plot.measureText(s0);
					
					
				}
				else if (c == '+' || c == '-' || c == '*' || c == '/' || c == '÷'
					|| c == '=' || c == ' ') {
					
					if (c == '*') {
						if (i > 0 && /[0-9]/.test(s[i-1]) && /[0-9]/.test(s[i+1])) {
							//对于乘号前后都是数字的情况，把乘号改成叉号
							c = ' \u00D7 ';
						}
						else {
							//对于代数式中，乘号改为点号
							c = ' \u00B7 ';
						}
					}
					
					//如果是运算符后的数被开根式，根号在运算符右边
					if (c == '-' || c == '/') {
						s0 += ' '+c+' ';
					}
					else {
						s0 += c;
					}
				
					if (bracketsLevel < 3) {
						xMem[bracketsLevel] = x+plot.measureText(s0);
					}						
				}
				else {
					s0 += c;				
				}				
				
			}
		}
		
		if (s0 != '') { //先把正常字符打印出
			if (r1 != 20) { //字体字号大小还在上标状态
				r1 = 20;
				this.setNormalFont();				
			}
			measure = plot.measureText(s0);
			plot.fillText(s0, x, y, measure);
			x += measure;
		}
		
		plot.restore();
	}


}
</span>

基本上到这里也是个底了吧，再向里塞功能怕要撑破肚皮了。

本节到此结束，欲知后事如何，请看下回分解。