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Mathematics, according to my favorite dictionary,
Merriam-Webster, is “the science of numbers and their operations,
interrelations, combinations, generalizations, and abstractions and of space
configurations and their structure, measurement, transformations, and
generalizations.”[i]
Or, to put it more simply, mathematics is “the science of
numbers, quantities, and shapes and the relationships between them.”[ii]
This second definition is the one I ask my students to learn.
Most of them have an even shorter definition of mathematics when I ask them. By
and large, my students define mathematics as “hard.”
Who can blame them for thinking mathematics is hard? Take a look
at all the different kinds of numbers there are for the students to learn. At
the level I teach, in the realm of all numbers, there are Real Numbers and
Imaginary Numbers.
Many people will look at my last sentence and shake their heads
at the idea of Imaginary Numbers. Such numbers do exist. Imaginary Numbers are
the product of a nonzero Real Number and the Imaginary Unit – usually the
square root of negative one.[iii]
Explaining the idea of Imaginary Numbers to seventh graders is a
challenging task. I usually save it for the end of the year, and then only for
the most advanced classes.
Within the realm of Real Numbers there are still enough different
types of numbers to confuse students. The two biggest sets of numbers we
discuss in seventh grade are Rational and Irrational Numbers. Since all sets of
numbers are, theoretically, infinite, it might be misleading to say the sets of
Rational and Irrational Numbers are the biggest, but from my students point of
view, they are the most intimidating.
The simple definition of Rational, according to Merriam-Webster,
is something based on facts or reason and not on emotions or feelings.[iv]
When dealing with Rational Numbers, my students often react with emotions and
feelings, and those feelings toward Rational Numbers are usually negative.
Most of us know Rational Numbers by their other name –
fractions. Technically, Rational Numbers are any number that can be expressed
in the form a/b where ‘a’ is equal to any Integer (all the whole numbers and
their opposites) and ‘b’ is equal to any nonzero Integer. When I was a student
I did not like fractions. Today, my students are not fond of fractions. I
imagine my students’ grandchildren will not be fond of fractions. But fractions,
and other forms of rational numbers such as ratios and decimals, must be faced,
learned, and mastered.
Just about the time students get used to the idea of Rational
Numbers, we hit them with Irrational Numbers. Irrational Numbers sound like
numbers that might not be able to use reason or good judgement.[v]
Irrational Numbers are actually defined as those numbers that cannot be written
in the form a/b as defined above. Most irrational numbers are the square roots
of numbers that are not perfect squares.
For instance, the square root of 4 is 2 because 2 times 2 equals
4. 4 is a perfect square. This doesn’t work for the square root of 5. No matter
how long you keep dividing out the square root of 5, the decimal will never
terminate, nor will it repeat. Thus, the square root of 5 cannot be expressed
in the form a/b. In other words, the square root of 5 cannot be expressed as a
ratio, so it is not rational. Therefore, the square root of 5 is irrational.
There are an infinite number of Irrational Numbers, as is true
with most types of numbers, which leaves a number of possibilities for future
articles.
As always, I remain,
The Exhausted Educator
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