Time estimate: 90 min.

2.9. Using the Math Class

Games would be boring if the same thing happened each time you played the game. Random numbers can be used in games to generate different outcomes each time the game is played. In Java, the Math.random() method to is used to generate a random number.

There are lots of mathematical methods in the Math class that you might want to use in your programs, like Math.pow(2,3) which calculates the 2 to the power of 3 which is 8.

These methods are in the Math class defined in the java.lang package. These are static methods which means you can call them by just using ClassName.methodName() without creating an object. This is why we can just say Math.random() instead of having to define an object of the class Math.

Note

Static methods (also called class methods) are called using the class name and the dot operator (.) followed by the method name. You do not need to create an object of the class to use them. You can use ClassName.methodName() or just methodName() if they are called from within the same class.

2.9.1. Mathematical Functions

The Math class contains the following methods that are in the AP CSA subset. There are more Math methods, outside of what you need on the AP exam, that you can find in the Math class Javadocs.

• int abs(int) : Returns the absolute value of an int value (which is the value of a number without its sign, for example Math.abs(-4) = 4).

• double abs(double) : Returns the absolute value of a double value.

• double pow(double, double) : Returns the value of the first parameter raised to the power of the second parameter.

• double sqrt(double) : Returns the positive square root of a double value.

• double random() : Returns a double value greater than or equal to 0.0 and less than 1.0 (not including 1.0!).

Note

All the Math methods that you may need to use or understand on the AP exam are listed in the AP CSA Java Quick Reference Sheet that you can use during the exam.

These Math methods are mathematical functions that compute new values from their arguments. You may be able to guess what abs, pow, and sqrt do, from their abbreviations.

Math.abs takes a single argument, either a double or an int and returns a value of the same type which is the absolute value of the argument. The absolute value is the positive value of the number without its sign or its distance from 0. So:

Math.abs(45);    // returns 45
Math.abs(-45);   // returns 45
Math.abs(33.3);  // returns 33.3
Math.abs(-33.3); // returns 33.3

Math.pow takes two argument, both doubles and returns a double which is the first argument raised to the power of the second argument.

Math.pow(2 , 3); // returns 8.0
Math.pow(10, 6); // returns 1000000.0
Math.pow(2, -3); // returns 0.125

Math.sqrt takes an double argument and returns a positive double value which is the square root of the argument. For example, the square root of 9 is 3 because 3 squared is 9.

Math.sqrt(9); // returns 3.0

Since these methods calculate and return a value, you need to use that value, for example in an assignment statement or in a print statement to see the result. For example:

System.out.println("The square root of 9 is " + Math.sqrt(9));

Coding Exercise

Try the Math methods below. Change the code so that it computes the absolute value of -4, the square root of 9, and 3 raised to the power of 2.

public class TryMath { public static void main(String[] args) { // TODO: Change the code below to compute // the absolute value of -4, // the square root of 9, // and 3 raised to the power of 2. System.out.println( Math.abs(-2) ); System.out.println( Math.sqrt(4) ); System.out.println( Math.pow(2, 3) ); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void testMain() throws IOException { String output = getMethodOutput("main"); String expect = "4\n3.0\n9.0\n"; boolean passed = getResults(expect, output, "Expected output from main after changing code"); assertTrue(passed); } }

Check Your Understanding

2-9-2: Knowing that Math.sqrt takes a single argument, Which of these are syntactically correct method calls to sqrt?

• Math.sqrt(2)

• ✅ This is a simple call to Math.sqrt with the argument 2.

• Math.sqrt()

• ❌ Math.sqrt takes one argument. This would be a correct call if it took no arguments.

• Math.sqrt(2, 4)

• ❌ Math.sqrt takes one argument. This would be a correct call if it took two arguments.

• Math.sqrt(2 + 3)

• ✅ The argument passed to Math.sqrt is the value of the expression 2 + 3, namely 5.

• Math.sqrt 2

• ❌ You must have parentheses around the arguments.

• Math.sqrt(Math.sqrt(2))

• ✅ The argument passed to Math.sqrt is the value of another call to Math.sqrt which is perfectly fine.

2-9-3: The distance between two numbers on the number line is defined as the absolute value of their difference. Their difference is just what you get when you subtract one from the other. For example, the distance from 0 to 3 is 3, the distance from -3 to 0 is 3, and the distance from -3 to 1 is 4.



Which of the following are correct expressions to compute the distance between the numbers a and b.

• Math.abs(a - b)

• ✅ a - b gives us the difference and Math.abs gives us the absolute value of that difference.

• Math.abs(a) - Math.abs(b)

• ❌ Consider the distance between -2 and 3. It should be five. What value would this expression produce in that case?

• Math.abs(a + b)

• ❌ We need to start with the difference between a and b, not their sum.

Coding Exercise

The distance between two numbers on a number line, as we discussed in the problem above, is defined as the absolute value of their difference. Their difference is just what you get when you subtract one from the other. For example, the distance from 0 to 3 is 3, the distance from -3 to 0 is 3, and the distance from -3 to 1 is 4.

Fill in the method distance below so it correctly computes the distance between two numbers a and b using subtraction and Math.abs.

public class DistanceCalculator { public static double distance(double a, double b) { // TODO: calculate the distance from a to b using subtraction and Math.abs. double distance = 0.0; return distance; } public static void main(String[] argv) { System.out.println("distance(13.5, 26.2) = " + distance(13.5, 26.2)); System.out.println("distance(26.2, 13.5) = " + distance(26.2, 13.5)); System.out.println(distance(13.5, 26.2) == distance(13.5, 26.2)); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void test1() { Object[] args = {10.5, 20.6}; String output = getMethodOutput("distance", args); String expect = "10.1"; boolean passed = getResults(expect, output, "distance(10.5, 20.6)"); assertTrue(passed); } @Test public void test2() { Object[] args = { -5.0, 2.4 }; String output = getMethodOutput("distance", args); String expect = "7.4"; boolean passed = getResults(expect, output, "distance(-5.0, 2.4)"); assertTrue(passed); } }

2.9.2. Random Numbers

The Math.random() method returns a double number greater than or equal to 0.0, and less than 1.0. When we talk about ranges of numbers sometimes we need to be precise about whether the ends of the range are part of the range. For example, Math.random returns a number between 0 and 1, but does that mean it can return exactly 0? Or exactly 1? As it turns out it can return 0 but never returns 1.

When we need to be precise about this we’d say that it returns a number between 0, inclusive, and 1, exclusive, meaning include 0 but exclude 1. Lots of ranges in Java are expressed this way, as you’ll see later on with an inclusive bottom and exclusive top.

Try the following code. Run it several times to see what it prints each time.

public class Test3 { public static void main(String[] args) { System.out.println(Math.random()); System.out.println(Math.random()); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void testMain() throws IOException { String output = getMethodOutput("main"); String expect = output; boolean passed = getResults(expect, output, "Expected output from main", true); assertTrue(passed); } }

Getting a number between 0, inclusive, and 1, exclusive, may not seem all that useful. But we can expand the range easily enough. To see how, imagine you had less than a dollar to your name and you wanted to be richer—you’d want to find a way to multiply your money. If you could invest every penny you had in something that would multiply your money by 1,000 then instead of having somewhere between $0 and $1, then you’d have somewhere between $0 (inclusive—if you started with $0) and $1,000 (exclusive, since if you had even a fraction of a penny less than $1 multiplying by 1,000 would still leave you just a bit shy of $1,000.) If the investment multiplied your original money by a million, you’d have between $0 and $1,000,000! (But never quite $1,000,000.)

Same trick applies to random numbers. The value Math.random returns is like the initial amount of money in your pocket, always a bit less than $1. If you multiply that value by any amount, it will stretch it into the range you want:

Coding Exercise

Try the following code. Run it several times to see what it prints each time. Did you ever see 0.0? How about 1.0?

public class StretchedRandom { public static void main(String[] args) { System.out.println(Math.random() * 10); System.out.println(Math.random() * 10); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void testMain() throws IOException { String output = getMethodOutput("main"); String expect = output; boolean passed = getResults(expect, output, "Expected output from main", true); assertTrue(passed); } }

You may have noticed that while the numbers generated were always in the range 0 to 10, all the numbers probably had a lot a digits after the decimal point. Often we want a random integer, with nothing after the decimal point. Easy enough—casting a double to an int will throw away any values after the decimal point. For example,

// rnd is an integer in the range 0-9 (from 0 up to 10).
int rnd = (int)(Math.random()*10);

Finally, what if we want a number in a range that doesn’t start with 0, say a number from 1 to 10 (including 10) instead of from 0 to 9 (including 9)? Since the size of the two ranges is the same, with ten numbers in each, all we need to do is shift from the range we’ve generated into the range we want. In other words, add the difference between the two ranges, 1 in this case.

// rnd is an integer in the range 1-10 (including 10).
int rnd = (int)(Math.random()*10) + 1;

Coding Exercise

Run the code below several times to see how the value changes each time. How could you change the code to return a random integer from 1 to 10? Modify the code and see if your answer is correct. Try removing the parentheses from around (Math.random() * 10) and run the code several times. What happens? The parentheses are necessary because (int) will cast the closest expression, and (int)Math.random() will always be 0 since anything after the decimal point is dropped.

public class Test4 { public static void main(String[] args) { System.out.println((int) (Math.random() * 10)); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void testContainsRange() throws IOException { String target = "+ 1"; boolean passed = checkCodeContains("Math.random in range 1 to 10", target); assertTrue(passed); } }

Note

• Math.random() returns a random number between 0.0-0.99.

• (int)(Math.random()*range) + min moves the random number into a range starting from a minimum number.

• The range is the (max number - min number + 1).

Here are some examples that move a random number into a specific range.

// Math.random() returns a random number between 0.0-0.99.
double rnd = Math.random();

// rnd1 is an integer in the range 0-9 (including 9).
int rnd1 = (int)(Math.random()*10);

// rnd2 is in the range 1-10 (including 10). The parentheses are necessary!
int rnd2 = (int)(Math.random()*10) + 1;

// rnd3 is in the range 5-10 (including 10). The range is 10-5+1 = 6.
int rnd3 = (int)(Math.random()*6) + 5;

// rnd4 is in the range -10 up to 9 (including 9). The range is doubled (9 - -10 + 1 = 20) and the minimum is -10.
int rnd4 = (int)(Math.random()*20) - 10;

So the general recipe for generating a random is to first stretch the value from Math.random() until it’s in the right size range by multiplying by the size of the range. Then if you want an integer value, cast to int to discard the part after the decimal point. Then shift the value up by adding the minimum value. The table below shows some applications of that general recipe.

Random recipes

Expression	Minimum (inclusive)	Maximum (exclusive)	Possible values
Math.random()	0.0	1.0	Over 9 quadrillion
Math.random() * 100	0.0	100.0	Over 9 quadrillion
(int)(Math.random() * 100)	0	100	100
(int)(Math.random() * 50) + 25	25	75	50
(int)(Math.random() * max)	0	max	max
(int)(Math.random() * range) + min	min	min + range	range
(int)(Math.random() * (max - min)) + min	min	max	max - min

Check your understanding

2-9-8: Which of the following would be true about 40% of the time?

• Math.random() < 0.4
• This is true about 40% of the time since Math.random returns a value from 0 to not quite 1.
• Math.random() > 0.4
• This will be true about 60% of the time.
• Math.random() == 0.4
• Do not use == with double values! Remember that Math.random can return any number between 0 and not quite 1 (about .99999999).

2-9-9: Which of the following would return a random number from 1 to 5 inclusive?

• ((int) (Math.random() * 5))
• This would be a number between 0 and 4.
• ((int) (Math.random() * 6))
• This would be a number between 0 and 5.
• ((int) (Math.random() * 5) + 1)
• The first part would return a number between 0 and 4 and when you add 1 you get a number from 1 to 5 inclusive.

2-9-10: Which of the following would return a random number from 0 to 10 inclusive?

• ((int) (Math.random() * 10))
• This would be a number between 0 and 9.
• ((int) (Math.random() * 11))
• This would be a number between 0 and 10.
• ((int) (Math.random() * 10) + 1)
• The first part would return a number between 0 and 9 and when you add 1 you get a number from 1 to 10 inclusive.

2-9-11: Which of the following would be true about 75% of the time?

• Math.random() < 0.25
• This is true about 25% of the time, since it will be a number from 0 to not quite 1.
• Math.random() > 0.25
• This is true about 75% of the time, since it will be a number from 0 to not quite 1.
• Math.random() == 0.25
• Do not use == with double values! Remember that Math.random can return any number between 0 and not quite 1 (about .99999999).

AP CSA Sample Problem

2-9-12: Which of the following statements assigns a random integer between 25 and 60, inclusive, to rn?

• int rn = (int) (Math.random() * 25) + 36;
• Remember that (int)(Math.random()*range) + min moves the random number into a range starting from a minimum number. We want the minimum number to be 25, but the minimum number here would be 36.
• int rn = (int) (Math.random() * 25) + 60;
• Remember that (int)(Math.random()*range) + min moves the random number into a range starting from a minimum number. We want the minimum number to be 25, but the minimum number here would be 60.
• int rn = (int) (Math.random() * 26) + 60;
• Remember that (int)(Math.random()*range) + min moves the random number into a range starting from a minimum number. Here the min is 25. We want the minimum number to be 25, but the minimum number here would be 60.
• int rn = (int) (Math.random() * 36) + 25;
• Yes, (int)(Math.random()*36) + 25 moves the random number into a range of 36 numbers starting from a minimum number 25 up to 60. The range is (max number - min number + 1) which is (60-25 +1) = 36.
• int rn = (int) (Math.random() * 60) + 25;
• This would give us random numbers from 25 to 85. Remember that you can compute the range you need with (max number - min number + 1).

2.9.3. Programming Challenge : Random Numbers

You may have a combination lock on your locker at school where you have to spin the dial to 3 separate numbers from 0 up to 40. What if you forgot your combination? Would you be able to guess it?

1. Write code that will generate 3 random integers from 0 up to 40 (but not including 40) using Math.random() in the Active Code window below. Run it a couple times to see it generate different numbers.

2. How many times would you need to run it to guess your combination correctly? Let’s have the code compute the number of permutations possible in your combination lock using Math.pow(number,exponent). For example, if you had 2 dials on your combination lock where each dial can be set to a digit from 0-9 (10 digits), there are 10² possible permutations. In Java, this would be written as Math.pow(10,2) which means 10 to the power of 2. If you start listing all the permutations possible, you can tell that there are 10² or 100 possible permutations for a 2 dial lock from 0-9.

00, 01, 02, 03, 04, 05, 06, 07, 08, 09
10, 11, 12, 13, 14, 15, 16, 17, 18, 19
...
90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Now what about the combination lock for this challenge? You will need to spin the dial 3 times: once to the right, once to the left, and once to the right to 3 different numbers from 0 up to 40 (not including 40). In general, the formula to use is NumbersPerDial^{numberOfDials}. Write this using the Math.pow() method in your code and save it into a variable and print out.

Complete the combination lock challenge below.

public class MathChallenge { public static void main(String[] args) { // 1. Use Math.random() to generate 3 integers from 0-40 (not // including 40) and print them out. // 2. Calculate the number of combinations to choose 3 numbers between // 0-40 (not including 40) using Math.pow() and print it out. // For example, Math.pow(10,2) is 10^2 and the number of permutations // to choose 2 numbers between 0-9. } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { @Test public void test1() { String output = getMethodOutput("main"); String[] lines = output.split("\\s+"); boolean passed = lines.length >= 2; passed = getResults( "2+ lines of output", lines.length + " lines of output", "Expected output", passed); assertTrue(passed); } @Test public void test2() { String output = getMethodOutput("main"); boolean passed = output.contains("64000"); passed = getResults("true", "" + passed, "Prints result of 40^3", passed); assertTrue(passed); } @Test public void test3() { String[] code = getCode().split("\n"); String expected = "Possible answers:\n(int) (Math.random() * 40)\n(int) (40 * Math.random())"; String actual = ""; int num = 0; for (int i = 0; i < code.length; i++) { if (code[i].contains("Math.random()") && code[i].contains("40")) { actual += code[i].trim() + "\n"; if (code[i].contains("(int)")) num++; } } boolean passed = num >= 3; passed = getResults( expected, actual, "Creates 3 random numbers from 0 to 40 (not inclusive)", passed); assertTrue(passed); } @Test public void test4() { String code = getCode(); int num = countOccurences(code, "Math.pow("); boolean passed = num >= 1; passed = getResults("1 or more", "" + num, "Calls to Math.pow(...)", passed); assertTrue(passed); } }

Here’s another challenge that is a lot of fun! Can you use random numbers to make dancing turtles? This idea was suggested by CSA teacher Zac Martin.

Complete the random numbers using Math.random() in the correct ranges to choose x, y coordinates and random color in the range of 0-255 for the turtle. Put on some music and watch your turtle dance!

import java.util.*; import java.awt.*; public class DancingTurtles { public static void main(String[] args) { World world = new World(500,400); Turtle yertle = new Turtle(world); // This is a loop that runs 10 times (you will learn to write loops in // Unit 4) for(int i = 1; i <= 10; i++) { // Can you choose a randomX between 0-500? // Can you adjust for the 20 pixel width of the turtle, // so it doesn't get cut off at the edges? // Move the range from 20 to 480. int randomX = 0; // Can you choose a randomY between 0-400? // Can you adjust for the 20 pixel height of the turtle, // so it doesn't get cut off at the edges? int randomY = 0; yertle.moveTo(randomX, randomY); yertle.turnRight(); // Can you choose a random red, green, and blue value between 0-255? int randomR = 0; int randomG = 0; int randomB = 0; yertle.setColor(new Color(randomR, randomG, randomB)); } // end of loop world.show(true); } } ==== import static org.junit.Assert.*; import org.junit.*; import java.io.*; public class RunestoneTests extends CodeTestHelper { public RunestoneTests() { super("DancingTurtles"); } @Test public void test1() { String code = getCode(); int numRandom = countOccurences(code, "Math.random()"); boolean passed = numRandom >= 5; passed = getResults("5+", "" + numRandom, "5+ calls to Math.random()", passed); assertTrue(passed); } @Test public void test2() { boolean passed = checkCodeContainsNoRegex( "Random numbers for 0-255 colors (256 values)", "Math.random() * 256"); assertTrue(passed); } }

2.9.4. Summary

• Static Math methods can be called using Math.method(); for each method.

• The following static Math methods are part of the Java Quick Reference:

  • int abs(int) : Returns the absolute value of an int value (which means no negatives).

  • double abs(double) : Returns the absolute value of a double value.

  • double pow(double, double) : Returns the value of the first parameter raised to the power of the second parameter.

  • double sqrt(double) : Returns the positive square root of a double value.

  • double random() : Returns a double value greater than or equal to 0.0 and less than 1.0 (not including 1.0)!

• The values returned from Math.random can be manipulated to produce a random int or double in a defined range.

• (int)(Math.random()*range) + min moves the random number into a range starting from a minimum number. The range is the (max number - min number + 1). For example, to get a number in the range of 5 to 10, use the range 10-5+1 = 6 and the min number 5: (int)(Math.random()*6) + 5.

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