This is what Theoni Pappas writes on page 150 of her Magic of Mathematics:
"Namely, the diagonal numbers 42; 25, 35 convert to 42 + (25/60) + (35/3600) = 42.42638889 while 30sqrt(2) = 42.42640687."
Oops -- well, we're on the last page of the first section, "Babylonians & Square Roots: What Were They Doing with Accurate Square Root Approximations?" We missed the rest of this section due to the Memorial Day weekend.
So let's just summarize the entire section. Pappas is describing the mathematics of the ancient Babylonians, who wrote on cuneiform tablets between 3000 and 200 BC.
The first key is the Babylonian numeral system itself. The number 1 was written as a down arrow, and then 2 through 9 were written as the same number of down arrows. Then the number 10 was written as a left arrow, and multiple left arrows denote 20, 30, 40, 50.
What makes Babylonian numerals unique is that they are based on sixty (sexagesimal), not ten. We still use the Babylonian system today when we consider minutes (1/60) and seconds (1/3600). Both Babylonian and modern minute-second systems are best described as six-on-ten rather than pure sixty, so, for example, the number 71 would be written as down, left, down. Pappas writes this with a comma as 1, 11, for one in the sixties place and eleven in the ones place. But I actually prefer 1:11, since the colon reminds us of the minute-second system. In any rate, the semicolon represents the "decimal" (or sexagesimal) point, so 1; 11 is actually 1 + 11/60.
And it's these fractions that are the focus of this chapter. Pappas writes that the Babylonians were able to estimate irrational square roots in the Babylonian system. They were able to estimate sqrt(2) to three sexagesimal places as 1; 24:51:10. This value is accurate to one part in 60^3 = 216000. Pappas calculates the decimal value as:
1 + 24/60 + 51/60^2 + 10/60^3 = 1 + 2/5 + 51/3600 + 1/21600 = 1.4142129+
which compares to sqrt(2) = 1.414213562.... She writes that the Babylonians might have found this value using a repetitive approximation and division method often used by the Greeks.
Last week's SBAC questions and our discussion of approximating square roots are on my mind. Of course we don't teach the approximation and division method any more because it's cumbersome to use during the current calculator age. But it was a great way to find square roots back before calculators existed. Notice that the Illinois State method gives sqrt(2) as 1 + 1/3, while the derivative method gives 1 + 1/2. Both of these are inaccurate, but notice that halfway between these is 1 + 5/12, which is fairly good. In sexagesimal, 1 + 5/12 is 1; 25, which is the same as the Babylonian rounded off to one place. (As 51 is more than half of 60, 1; 24:51 rounds up to 1; 25.)
The Pappas approximation of 30sqrt(2) would have been a cinch for the Babylonians to find once they had sqrt(2), since 30 is half of the base 60. They only needed to multiply by 60 (one-place shift) and then divide by two.
By the way, the numbers sqrt(2) and 30sqrt(2) were found drawn on clay tablets near the diagonal of a square. This implies that the Babylonians knew that the diagonal of a square is sqrt(2) times the length -- and this might have been centuries before Pythagoras.
Speaking of irrationals, Question 5 of the SBAC Practice Exam is on irrational numbers:
5. Determine for each number whether it is a rational or an irrational number.
-- -1 2/3
Well, -1 2/3 and 0.423 are clearly rational. For 1/sqrt(9), we must recall that sqrt(9) = 3, so that this equals the rational number 1/3. Only sqrt(17) is irrational, since 17 is not a perfect square. So the answer must be:
rational - irrational - rational - rational
By the way, a good Babylonian approximation to sqrt(17) is 4; 7:23:11, to three sexagesimal places.
Question 6 of the SBAC Practice Exam is on the equation of a line:
6. Consider the line shown on the graph.
Enter the equation of the line in the form y = mx where m is the slope.
Naturally, the line passes through the origin since there is no b in the equation. This is another graph where the slope appears to be 1 until we look at the scale on each axis. We observe that the y-scale is 5 while the x-scale is 10. So the slope is 5/10 = 1/2, and so the equation is y = (1/2)x.
SBAC Practice Exam Question 3
Common Core Standard:
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
SBAC Practice Exam Question 4
Common Core Standard:
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Commentary: The first question confused my eighth graders, until they are reminded first what an irrational number is and second what sqrt(9) is. For the second question, slope should have been still familiar for the students after having seen last week's worksheet. For my students, entering y = (1/2)x on the computer is a little tricky, but fortunately, they've had practice entering equations with fractions on IXL.