Real Numbers, From The Ground Up
What is a real number, really? What kinds of numbers make them up? What do they represent?
Natural Numbers: Counting
Starting at the most concrete level, a number is a count of how many objects there are. There are six bananas in my fruit basket. I have two children. There are eleven players in a cricket team. And so on. This notion of a number as a count is, for most of us, our first encounter with numbers. Such numbers are vividly reflected in the physical world. They are apparent even to some animals (NYT article).
Such a number that is a count of things is known as a natural number. It can be represented quite literally with tally marks equal to the count. So three tally marks represent a count of three objects.
Positional Number Systems: Representation
The tally marks representation of natural numbers will rapidly become unwieldy as we start counting larger sets of things (e.g. the number of people in the United States). So it's convenient to use a more compact notation to represent counts. One system for such a notation is called a positional number system.
What is a positional number system? When we first begin counting, as little children, we use our fingers (digits) to count things up to ten. We learn about the symbols (digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and understand that the symbol '3' represents us holding up three fingers when counting three apples. Then, when we increase in sophistication to be able to count larger sets, we don't keep inventing new symbols for the larger counts. It would be impossible to remember a unique symbol for each unique count. Instead, we represent each count using just the ten symbols (digits) by forming a "word" of digits. Each digit in the word has a place value (ones for the rightmost digit, tens for the next one on the left, and so on). So the number 420 denotes 4 hundreds, 2 tens, and 0 ones. This is the familiar decimal or base ten number system that we all use every day.
There's nothing special about base ten, other than the fact that we have ten fingers. A natural number can just as well be represented in number systems of different bases. A base b number system has b digits and the place values from right to left or ones, 'b's, 'b²'s, etc.
For example, in a base 8 number system (called octal) the digits are 0, 1, ..., 7. The number 420₈ (the subscript denotes base 8) represents 4 sixty-fours, 2 eights, and 0 ones or 272 in base 10.
Computers use a base 2 number system (called binary) in which numbers are represented as words of 0s and 1s. The binary number 101₂ represents 1 four, 0 twos, and 1 one or 5 in base 10.
The base can also be larger than ten. For example, the base 16 number system (called hexadecimal) has the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f. The digit ‘a’ has the decimal value 10, ‘b’ has value 11, and so on. The number 6a₁₆ represents 6 sixteens and 10 ones or 106 in base 10.
If you're a kid and desire to project yourself as older than you are then feel free to use a base smaller than 10 to inflate your age. For example, if you're 11 years old, then you can pretend to be a teenager by stating your age in base 8 as 13. Or, if you would rather be in your thirties when you're actually 45, would it be a lie if you say that you're 39 (in base 12)?
Whole Numbers: Emergence of 0
Now that we’ve thought carefully through natural numbers that represent counts of things in the real world, and how to compactly represent such numbers using a positional number system, let's turn our attention to doing some arithmetic using these numbers. If you have 6 pencils and you buy 5 more pencils, you end up with 11 pencils. This physical addition of 5 pencils to 6 pencils is represented in arithmetic as 6 + 5 = 11. If you give away 3 of the 11 pencils you are left with 8 pencils. This physical subtraction of 3 pencils from 11 pencils is represented in arithmetic as 11 - 3 = 8.
There is a geometric (pictorial) representation of natural numbers that allows us to represent them as discrete points on a number line without the need to use any specific number system. Then, the addition of 5 to 6 means starting at the '6' point on the number line and moving '5' points to its right. Similarly subtracting 3 from 11 means starting at the '11' point on the number line and moving '3' points to its left.
So what happens if we take away (subtract) everything we started out with, say subtract 5 from 5? Now natural numbers start diverging from concretely countable reality and we take our first step into the abstract. As we count leftward from 5, after four hops we reach the smallest natural number, 1, on the number line. Then the fifth leftward hop forces us to extend the number line leftward by one point. This point is the number zero (0). Zero does not have a clear representation in reality. Yet we are forced to accept it if we are to make sense of subtracting a natural number from itself.
The set of natural numbers, along with 0, is called whole numbers. Zero emerges when a natural number is subtracted from itself.
Integers: Emergence of Negatives
The same subtraction argument that gave rise to 0 on the number line gives rise to negative numbers. What happens if we take away (subtract) more than we started out with, say subtract 8 from 5? Well, we start at 5 on the number line. After 5 hops we reach 0. But we still have 3 leftward hops to go. So we extend the number line leftward from 0 and add the points -1, -2, -3, and so on. In addition to whole numbers we now have negative numbers. Such negative numbers don't have a clear representation in reality. Yet we are forced to accept them if we are to make sense of subtracting a larger whole number from a smaller whole number.
Whole numbers (0, 1, 2, ...) and negative numbers (-1, -2, ...) are together are called integers. Integers are closed under addition and subtraction i.e. when two integers are added or subtracted the result is an integer. This means we can stay within the world of integers and add and subtract them willy-nilly without worrying about the question “is there something else”. But that question will rear its head in the next section.
Rational Numbers: Between Two Integers
What do the points between two integers on the number line denote? Is that just dead space or is there something there?
To answer these questions, consider the arithmetic operation of multiplication. A real-world interpretation of multiplication is repeated addition. So, if there are 4 kids and each kid has 3 apples, there are 4 x 3 = 4 + 4 + 4 = 12 apples in all. This example still keeps us in the space of integers, nothing new to see so far. But just like zero and negative numbers emerged from the inverse of addition (subtraction), something new emerges from the inverse of multiplication (division). Here's a real-world example. Divide an apple pie into 8 equal pieces and serve yourself 3 of those pieces. You have just served yourself a quantity of the pie that is represented on the number line as a point somewhere between 0 and 1. The location of the point is obtained by dividing the interval between 0 and 1 into 8 equal parts (for the 8 slices the whole pie was divided into) and picking the point at which the 3rd part ends. This point is represented in base 10 as the fraction 3/8.
Integers and fractions together are called rational numbers. Rational numbers are closed under all arithmetic operations (addition, subtraction, multiplication, division) i.e. a rational number plus another rational number is a rational number. And the same holds for subtraction, multiplication, and division.
A rational number can be expressed as p/q where p is an integer and q is a natural number. When q = 1, we get natural numbers when p > 0, whole numbers when p ≥ 0, and integers otherwise. When q ≠ 1 we get simple fractions (as long as p is not a multiple of q).
Irrational Numbers: Filling In The Number Line
In our numbers journey so far we have visited natural numbers (1, 2, 3, ...), whole numbers (natural numbers and 0), integers (whole numbers and negatives), and fractions (points between integers obtained by dividing the interval between integers into a fixed number of equal parts). All these numbers together are known as rational numbers. And we can perform the arithmetic operations of addition, subtraction, multiplication, division on rational numbers and get rational numbers as answers. Do rational numbers account for all possible numbers? Are there points on the number line not covered by rational numbers?
To answer the question of whether there's anything beyond rational numbers on the number line, it's useful to look at the decimal expansion of non-integers. First, consider a non-integer with a finite number of digits after the decimal point e.g. 0.31. Such a number is clearly the rational number 31/100. In general, a decimal expansion with a finite number of digits is a rational number. Nothing new to see here.
Next, consider a non-integer with an infinitely repeating pattern of digits after the decimal point e.g. 0.531313131... Such a number turns out to be a rational number as well, though it's a bit harder to see that. Let A = 0.531313131... Then 10A = 5.31313131... and 1000A = 531.31313131... So 1000A - 10A = 526 => A = 526/990. In general, a decimal expansion with an infinitely repeating pattern of digits after the decimal point is a rational number. Again, nothing new to see here.
Finally, consider a non-integer with an infinite number of digits after the decimal point but with no infinitely repeating pattern. We can imagine constructing such a number by picking a random digit for each place after the decimal point (though such a process will never terminate because there are infinitely many places after the decimal point, but we get the idea). There are many, many such numbers. Infinitely many. And such a number cannot be expressed as a rational number. An intuitive way to see that is that such a number has infinite information (since there are infinite digits with no infinitely repeating pattern) but a rational number has finite information (two finite integers). Such numbers with infinite non-repeating digits after the decimal point are called irrational numbers.
Irrational numbers are not closed under arithmetic. For example, an irrational number minus itself would give the rational number 0.
Real Numbers
Rational numbers and irrational numbers together make up real numbers. Now that we have rational and irrational numbers filling the intervals between integers, are we done? Are there any other gaps on the number line? No! We are done. We have fully covered the number line. Every point has been accounted for. There are no remaining decimal representations of numbers to consider.
In “Number Of Numbers: Infinite Weirdness”, we explore how many numbers there are of each kind (spoiler alert: infinite), how some infinities are the same as others (sometimes counter-intuitively), and how some infinities are larger than other infinities (mind-blowingly).
