Symbols and Science

In Memoriam Andrew Gleason

Cardano, Sponge Cake, and Notation

Cardano’s famous book ”Ars Magna” (1545) marked the end of the
medieval period in mathematics and triggered the development of
modern mathematical notation. The famous formula for the cubic , as
presented in the ”Ars Magna”, resembles more a culinary recipe than
a modern formula. The switchover has implications for the teaching of
mathematics.

One view of algebra : quadratic equations

The quadratic formula

A number x satisfies

x² + bx + c = 0

if and only if

Factoring

If r + s = -b and rs = c, then

x² + bx + c = (x - r )(x - s)

and

x² + bx + c =0if and only if x = r or x = s:

Another view of algebra: understanding functions

H (daylight in hours)

Figure: Sunlight in Madrid

The graph shows the number of hours of daylight in Madrid for one
month.
Why does the graph look linear ?
Estimate and interpret the slope of the line .
What month does the graph show?

What symbolic skills do modern students need?

Manipulative skills?
Graphical and numerical understanding ?
Symbol sense?

Al-Khwarismi, Hisab al-jabr w’al-muqabala, 830

what is the square which combined with ten of its
roots will give a sum total of 39? The manner of solving
this type of equation is to take one-half of the roots just
mentioned. . . . Therefore take 5, which multiplied by
itself gives 25, an amount which you add to 39 giving
64. Having taken then the square root of this which is
8, subtract from it half the roots, 5 leaving 3.

x² + 10x = 39
x² + 10x + 25 = 39 + 25 = 64
x + 5 = 8
x = 3

The quadratic formula in the 17th century

What is going on here?

Viete’s formulae and the quadratic formula

x² + bx + c = 0

then let

Exercise

Give an explanation, purely in terms of the structure of the
expressions, of why these two numbers satisfy
r + s = -b and rs = c:

Answer

When you add r and s, the plus and minus signs cancel.

If

x² + bx + c = 0

then let

Answer

When you add r and s, the plus and minus signs cancel.

If

x² + bx + c = 0

then let

Answer

When you add r and s, the plus and minus signs cancel.
When you multiply r and s, you get the difference of two squares in
the numerator ,