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

or

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.

Practice with square roots for homework due July 6.

The editor for Blogger doesn't have a square root sign available, so I will use sqrt(2) to signify the square root of 2, for example.



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 = __________

Here are some fractions with the square root in the denominator.  Write them in standard form and simplify.

3) 20/sqrt(10)

4) 15/sqrt(6)


Find the distance between the two given points using the formula Distance = sqrt((x1 - x2)² + (y1 - y2)²). Write the number as a square root in simplified form and give the approximation to the nearest thousandth.

5)  (3, 7) and (-2, 6)

6) (3, 1) and (-5, -9)

Answers in the comments.

Square root practice.

The square root sign is not available with the editor for Blogger (not an extended ASCII symbol is the nerdy explanation), so I will be writing sqrt(n) to indicate the square root of n.

The Pythagorean Theorem tells us that a² + b² = c², where a and b are the short sides of a right triangle, called the "legs", and c is the long side opposite the right angle, known as the "hypotenuse". This means since the square of the hypotenuse is equal to a² + b², the hypotneuse itself is equal to sqrt(a² + b²). What this means is we can find the square root of any number that is the sum of two perfect squares. Here are some examples.

If the legs are 1 and 1, then 1² + 1² = 2 = c², so c = sqrt(2)
If the legs are 1 and 2, then 1² + 2² = 5 = c², so c = sqrt(5)
If the legs are 1 and 3, then 1² + 3² = 10 = c², so c = sqrt(10)

We can also re-arrange the equation to a² = c² - b², which means we can find the square root of any number that can be written as the difference of two squares.

If the hypotenuse is 2 and one of the legs is 1, then 2² - 1² = 3 = a², so a = sqrt(3)
If the hypotenuse is 3 and one of the legs is 1, then 3² - 1² = 8 = a², so a = sqrt(8)
If the hypotenuse is 5 and one of the legs is 2, then 5² - 2² = 21 = a², so a = sqrt(21)

Here are some practice problems.

1) Make a right triangle that will have two whole number side lengths and one side with length of sqrt(7).

2) Make a right triangle that will have two whole number side lengths and one side with length of sqrt(18).

3) Make TWO different right triangles that will have two whole number side lengths and one side with length of sqrt(20).

4) Make TWO different right triangles that will have two whole number side lengths and one side with length of sqrt(50).

Answers in the comments.