Mathematics is much more than numbers. It includes shapes, logic, symbols, spaces, and broad practices like critical thinking and attending to precision, along with applications far and wide in everything from physics to physical education. But ask someone what math is, and you will almost always hear an answer involving numbers. They are often our introduction into math and a salient way that math is found in the real world.
So what is a number?
It is not an easy question to answer. It was not always known, for example, how to write and perform arithmetic with zero or negative quantities. The notion of number has evolved over millennia and has, at least apocryphally, cost one ancient mathematician his life.
Natural, Whole, and Integer Numbers
The most common numbers that we encounter—in everything from speed limits to serial numbers—are natural numbers. These are the counting numbers that start with 1, 2, and 3, and go on forever. If we start counting from 0 instead, the set of numbers are instead called whole numbers.
While these are standard terms, this is also a chance to share how math is ultimately a human endeavor. Different people may give different names to these sets, even sometimes reversing which one they call natural and which one they call whole! Open it up to your students: what would they call the set of numbers 1, 2, 3...? What new name would they give it if they included 0?
The integer numbers (or simply integers) extend whole numbers to their opposites too: ...–3, –2, –1, 0, 1, 2, 3.... Notice that 0 is the only number whose opposite is itself.
Rational Numbers and More
Expanding the concept of number further brings us to rational numbers. The name has nothing to do with the numbers being sensible, although it opens up a chance to discuss ELA in math class and show how one word can have many different meanings in a language and the importance of being precise with language in mathematics. Rather, the word rational is related to the word found within the first five letters: ratio.
A rational number is any number that can be written as the ratio of two integers, such as \(\frac{1}{2}\), \(\frac{783}{62,450}\) or \(\frac{-25}{5}\). Note that while ratios can always be expressed as fractions, they can appear in different ways, too. For example, \(\frac{3}{1}\) is usually written as simply \(3\), the fraction \(\frac{1}{4}\) often appears as \(0.25\), and one can write \(-\frac{1}{9}\) as the repeating decimal \(-0.111\)....
Any number that cannot be written as a rational number is, logically enough, called an irrational number. And the entire category of all of these numbers, or in other words, all numbers that can be shown on a number line, are called real numbers. The hierarchy of real numbers looks something like this:
An important property that applies to real, rational, and irrational numbers is the density property. It says that between any two real (or rational or irrational) numbers, there is always another real (or rational or irrational) number. For example, between 0.4588 and 0.4589 exists the number 0.45887, along with infinitely many others. And thus, here are all the possible real numbers:
Real Numbers: Rational
Key standard: Understand a rational number as a ratio of two integers and point on a number line. (Grade 6)
Rational Numbers: Any number that can be written as a ratio (or fraction) of two integers is a rational number. It is common for students to ask, are fractions rational numbers? The answer is yes, but fractions make up a large category that also includes integers, terminating decimals, repeating decimals, and fractions.
- An integer can be written as a fraction by giving it a denominator of one, so any integer is a rational number.
\(6=\frac{6}{1}\)
\(0=\frac{0}{1}\)
\(-4=\frac{-4}{1}\) or \(\frac{4}{-1}\) or \(-\frac{4}{1}\) - A terminating decimal can be written as a fraction by using properties of place value. For example, 3.75 = three and seventy-five hundredths or \(3\frac{75}{100}\), which is equal to the improper fraction \(\frac{375}{100}\).
- A repeating decimal can always be written as a fraction using algebraic methods that are beyond the scope of this article. However, it is important to recognize that any decimal with one or more digits that repeats forever, for example \(2.111\)... (which can be written as \(2.\overline{1}\)) or \(0.890890890\)... (or \(0.\overline{890}\)), is a rational number. A common question is "are repeating decimals rational numbers?" The answer is yes!
Integers: The counting numbers (1, 2, 3,...), their opposites (–1, –2, –3,...), and 0 are integers. A common error for students in Grades 6–8 is to assume that the integers refer to negative numbers. Similarly, many students wonder, are decimals integers? This is only true when the decimal ends in ".000...," as in 3.000..., which is equal to 3. (Technically it is also true when a decimal ends in ".999..." since 0.999... = 1. This doesn't come up particularly often, but the number 3 can in fact be written as 2.999....)
Whole Numbers: Zero and the positive integers are the whole numbers.
Natural Numbers: Also called the counting numbers, this set includes all of the whole numbers except zero (1, 2, 3,...).
Real Numbers: Irrational
Key standard: Know that there are numbers that there are not rational. (Grade 8)
Irrational Numbers: Any real number that cannot be written in fraction form is an irrational number. These numbers include non-terminating, non-repeating decimals, for example \(\pi\), 0.45445544455544445555..., or \(\sqrt{2}\). Any square root that is not a perfect root is an irrational number. For example, \(\sqrt{1}\) and \(\sqrt{4}\) are rational because \(\sqrt{1}=1\) and \(\sqrt{4}=2\), but \(\sqrt{2}\) and \(\sqrt{3}\) are irrational. All four of these numbers do name points on the number line, but they cannot all be written as integer ratios.
Non-Real Numbers
So we've gone through all real numbers. Are there other types of numbers? For the inquiring student, the answer is a resounding YES! High school students generally learn about complex numbers, or numbers that have a real part and an imaginary part. They look like \(3+2i\) or \(\sqrt{3}i\) and provide solutions to equations like \(x^2+3=0\) (whose solution is \(\pm\sqrt{3}i\)).
In some sense, complex numbers mark the "end" of numbers, although mathematicians are always imagining new ways to describe and represent numbers. Numbers can also be abstracted in a variety of ways, including mathematical objects like matrices and sets. Encourage your students to be mathematicians! How would they describe a number that isn't among the types of numbers shown here? Why might a scientist or mathematician try to do this?
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