64
R. Barrett and P. P. Delsanto
Problems of the type: what is the price of one cookie, if it costs $12 to buy
four ? are common in commerce. The answer is, of course, $3. Suppose now
that we replace “cookie” in the above question with some other object, e.g.
liquorice stick. The answer is still the same, i.e. $3, irrespective of what object
we choose. We can show this independence of the nature of the object by
writing the question in the form: what is the price of x, if it costs $12 to buy
4x ?, or even more succinctly by: Find x if 4x = 12. The answer is x = 3 (i.e.
12/4) dollars, if our unit of currency happens to be the dollar. What we have
introduced above is the concept of an equation (where two quantities, one
of which is unknown, are located on opposing sides of an equals sign. The
rules for solving the equation (i.e. finding the value of the unknown x ) are
contained in elementary algebra.
So far, so good, and for the moment our integer number world seems
adequate for our purposes. However, what happens if four cookies cost $10?
(i.e. 4x = 10). We easily see that there is no integer in our number world
to tell us how much one cookie costs. We do not need a calculator to find
that x = 10/4 = $2.50. But 2.50 is no longer an integer. In other words, the
number world of integers is too limited to include the solution of our new
equation. To solve it we must go into a much wider number world, the world
of “rational ” numbers, which includes integers and fractions, such as 10/4,
or 2.50.
Let us now assume that our problem is even more complicated, and we are
asked to find a number (again let us call it x ) such that when it is multiplied
by itself (or squared ) we obtain the value 4. We can write this in the form:
x *x = 4, or more concisely: x 2 = 4. Solving this equation presents no great
problem, and we see that x = 2, because the square of 2 is 4, or alternatively,
the square root of 4 is 2. (Another possible solution to this equation is x = −
2).
Now, suppose we ask ourselves what is the square root of 2? This is a
perfectly legitimate question. Imagine that we construct a triangle having two
sides of unit length with a right angle (i.e. an angle of 90°) between them,
then the length of the third side (i.e. the hypotenuse) is equal to the square
root of 2, which turns out to be 1.414213562373095… (See Fig. 4.1.) Here
we have arbitrarily truncated the number of decimal places to 15. In fact,
there is no limit to the number of decimal places, and we could have filled the
rest of the book with them. If we had, we would not have found any repeated
R. Barrett and P. P. Delsanto
Problems of the type: what is the price of one cookie, if it costs $12 to buy
four ? are common in commerce. The answer is, of course, $3. Suppose now
that we replace “cookie” in the above question with some other object, e.g.
liquorice stick. The answer is still the same, i.e. $3, irrespective of what object
we choose. We can show this independence of the nature of the object by
writing the question in the form: what is the price of x, if it costs $12 to buy
4x ?, or even more succinctly by: Find x if 4x = 12. The answer is x = 3 (i.e.
12/4) dollars, if our unit of currency happens to be the dollar. What we have
introduced above is the concept of an equation (where two quantities, one
of which is unknown, are located on opposing sides of an equals sign. The
rules for solving the equation (i.e. finding the value of the unknown x ) are
contained in elementary algebra.
So far, so good, and for the moment our integer number world seems
adequate for our purposes. However, what happens if four cookies cost $10?
(i.e. 4x = 10). We easily see that there is no integer in our number world
to tell us how much one cookie costs. We do not need a calculator to find
that x = 10/4 = $2.50. But 2.50 is no longer an integer. In other words, the
number world of integers is too limited to include the solution of our new
equation. To solve it we must go into a much wider number world, the world
of “rational ” numbers, which includes integers and fractions, such as 10/4,
or 2.50.
Let us now assume that our problem is even more complicated, and we are
asked to find a number (again let us call it x ) such that when it is multiplied
by itself (or squared ) we obtain the value 4. We can write this in the form:
x *x = 4, or more concisely: x 2 = 4. Solving this equation presents no great
problem, and we see that x = 2, because the square of 2 is 4, or alternatively,
the square root of 4 is 2. (Another possible solution to this equation is x = −
2).
Now, suppose we ask ourselves what is the square root of 2? This is a
perfectly legitimate question. Imagine that we construct a triangle having two
sides of unit length with a right angle (i.e. an angle of 90°) between them,
then the length of the third side (i.e. the hypotenuse) is equal to the square
root of 2, which turns out to be 1.414213562373095… (See Fig. 4.1.) Here
we have arbitrarily truncated the number of decimal places to 15. In fact,
there is no limit to the number of decimal places, and we could have filled the
rest of the book with them. If we had, we would not have found any repeated