Tuesday, June 26, 2012

Square roots

Squares and square roots are what we call inverse operations.  If we take a number x and square it, the value is x². If we take the square root of x², we get the value x. A square, of course, is a geometric shape with four equal sides and four 90° angles. If we know the length of the side and call it s, the area is s², measured in square units. If instead we know the area is a number we call A square units, the side length for all the sides is the square root of A, which I will write on the blog as sqrt(A).

I will repeat this many times in class. When we are doing these kinds of problems, squares give us answers that are measured in square units which represent surface area, but square roots produce answers that are measure in regular units, which is to say lengths, one dimensional measure.

The Pythagorean Theorem is one of the central ideas in math, and it deals with the sides of a right triangle, which means a triangle that has one right angle. (A right angle measures 90°.) It is standard to name the three sides a, b and c, where c is the long side, always opposite the right angle. We call the long side the hypotenuse and the short sides are the legs.  The formula connected to the theorem is

a² + b² = c²

We can re-arrange the letters in the two following ways

a²  = c² - b²


b² = c² - a²

What this means is that though there are three numbers in the formula, if you have the values of any two of them, you can find the value of the third. In math, we say this set of numbers has two degrees of freedom.  ("Degrees of freedom" is not the same as "degrees", the units we use to measure angles.) You are "free" to use any two numbers for the lengths a and b, but once those are chosen, we can find c by using the formula and the square root operation, which means c is determined by a and b and can't be just any old number.  Likewise, you are "free" to pick and two numbers and call them b and c (or a and c), provided that c is bigger than the other number.  But once you have chosen two numbers out of the three, the third number is forced to be a specific value by using the formula.

A number is a perfect square or a square number if it is the square of a whole number.  Here are the perfect squares less than or equal to 100.

0² = 0  The square root of 0 is 0.
1² = 1  The square root of 1 is 1.
2² = 4  The square root of 4 is 2.
3² = 9  The square root of 9 is 3.
4² = 16  The square root of 16 is 4.
5² = 25  The square root of 25 is 5.
6² = 36  The square root of 36 is 6.
7² = 49  The square root of 49 is 7.
8² = 64  The square root of 64 is 8.
9² = 81  The square root of 81 is 9.
10² = 100  The square root of 100 is 10.

If a number is not a perfect square, its square root will not be a whole number. In fact, it will be an irrational number, a number that cannot be written as a fraction p/q. This means that as a decimal representation, we never get a pattern of digits that repeats forever. For example, the square root of 2 is represented on your calculator as 1.414213562... Any representation of a square root that has a finite number of digits is only an approximation.  It is common to round square roots to three places after the decimal, which is to say the nearest thousandth. When I write an approximation on the blog, I will use the symbols "~=".

Example: sqrt(2) ~= 1.414

The square root of 2 is the number whose square is exactly 2. but 1.414 is not the exact value.

1.414² = 1.999396

Besides approximating square roots, which we will be doing with calculators, there is also the idea of simplifying square roots, which will involve factoring a number down to its prime factorization.

Here are some practice problems for simplifying square roots, a method we learned today in class.

Here are practice problems for using the Pythagorean Theorem to find the length of the third side of a right triangle if you already know the lengths of the other two sides.

Here are some more practice problems using the Pythagorean Theorem.

Here are two sides of a right triangle.  In these problems, c is always the hypotenuse and a and b are the short sides, also called the legs.  Find the missing side using the Pythagorean Theorem, write it as a square root in simplified form and give the approximation to the nearest thousandth.

1)  a = 7, b = 6, c = _________

2) b = 6, c = 7, a = __________

3)  a = 8, b = 6, c = _________

4) a = 6, c = 8, b = __________

The answers are in the comments.

1 comment:

Prof. Hubbard said...

1) a = 7, b = 6, c = _________

7² + 6² = 49 + 36 = 85

If c² = 85, c = sqrt(85) ~= 9.220.

Because 85 = 17 x 5 as its prime factorization, we cannot simplify the square root any farther.

c = sqrt(85) is the simplest form.

2) b = 6, c = 7, a = __________

7² - 6² = 49 - 36 = 13

If a² = 13, a = sqrt(13) ~= 3.606.

Because 13 is prime, we cannot simplify the square root any farther.

a = sqrt(13) is the simplest form.

3) a = 8, b = 6, c = _________

8² + 6² = 64 + 36 = 100

If c² = 100, c = sqrt(100) = 10.

Because 100 is a perfect square, the whole number answer is the simple form. What this means is that writing sqrt(100) is NOT the simple form, 10 is. If you leave a perfect square in square root form as a final answer, it will always be marked as at least partially wrong.

c = 10 is the simplest form.

4) a = 6, c = 8, b = __________

8² - 6² = 64 - 36 = 28

If c² = 28, c = sqrt(28) = 5.292.

28 = 2 x 2 x 7, so we have a perfect square in the prime factorization and we can perform a jailbreak, as Mrs. Kruger called it.

c = 2sqrt(7) is the simplest form.