accompanying illustration, the 6 on line 5
is the sum of the pair of 3’s above; the next
line is 1, 10 (1 9), 45 (9 36), 120 (36
84), and so on. Although the triangle was
known to both the Chinese and the Arab
cultures for several hundred years before,
it was named after the person who brought
it to the forefront of mathematics: French
mathematician Blaise Pascal (1623–1662).
(For more information about Pascal, see
“History of Mathematics.”)
How can one visualize Pascal’s
triangleusing algebra?
One way of looking at Pascal’s triangle is
through its connection to algebra. For example, expand (or remove the brackets
around) the expression (1 x)^2 (1 x)(1 x) 1 2x x^2. The same can be
done with a cube; for example, (1 x)^3 (1 x)(1 x)(1 x) (1 x)(1 2x
x^2 ) 1 3x 3x^2 x^3 ; and even the expression (1 x)^4 , which equals 1 4x 6x^2
4x^3 x^4.
The coefficients (the numbers in front of the x’s) in the results are the connection.
For the first example, the coefficients are 1, 2, 1; for the second one, 1, 3, 3, 1; and for
the last expression, the coefficients are 1, 4, 6, 4, 1. These, of course, are the third,
fourth, and fifth lines from Pascal’s triangle.
What are some “life questions”you can figure out using math?
There are many questions you can explore about your own body and age with mathe-
matics. For example, approximately how many Sunday nights can you expect to sleep
until you are 100 years old? Just take 100 years, minus your current age, and multiply
that result by 52 (weeks in a year with a Sunday). For example, if you are 25, the
answer would be (100 25) 52 3,900. How many of those will be good night’s
sleep is up to you.
To calculate the number of times your heart has beaten since you were born, you
need the help of a watch or clock. First, time your heartbeats per minute (to find out
how to count your pulse, see “Everyday Math”); then multiply beats per minute 60
minutes (in an hour) 24 hours (in a day) 365.25 days (in a year) your age. For
example, 72 heart beats 60 24 365.25 a person who is 30 1,136,073,600
beats since the person was born. Of course, this is an approximation, since the heart
usually beats slower at night, and it speeds up when you see the bill for your latest
car repair. 439
RECREATIONAL MATH
In Pascal’s triangle, each number is equal to the two
numbers directly above it.