Numbers

by Richard Reid

Mathematics began with numbers. Later it branched into topics on shapes and logic, but to start with it began with numbers. Your own mathematical experience probably started by counting on your fingers: 1, 2, 3, 4, 5, and so on. Certainly if you count cards or wish to count cards, then you must have experience with numbers.

This section outlines the terminology of numbers so that later when we use the terms one can know what we are talking about.

Natural Numbers:
The numbers you count on your fingers, such as 1, 2, 3, and so on are called natural numbers or in some books, counting numbers. These numbers are used for quantifying the things in the world around you. In blackjack, natural numbers are used to specify the number of decks in play, or the number of players at a table, or even the number of blackjack tables in a casino.

At first, it may seem like the natural numbers are the only numbers needed. You can add any two natural numbers, or you could multiply any two natural numbers and the result will always be a natural number. You learned your addition and multiplication tables in school and for a while did not have to worry about any "unnatural" numbers.

Whole Numbers:
However, when you began to learn subtraction, you quickly found that the natural numbers were not enough. For example, if as soon as you want to subtract a number from itself, like the number 2 from the number 2, you no longer get a number you can count on your fingers. Therefore, cases like 2 - 2 = 0, required that the number zero be added to the set of natural numbers. Hence, the set of whole numbers was created. A "whole number" is a digit from 0 to 9, or a combination of digits, such as 491, or 1,305,267. The set of whole numbers can be shown as {0, 1, 2, 3, 4 . . . }.

Integers:
Yet even the set of whole numbers was not enough to cover all the results of subracting two numbers. For example, "What is the result of "2 take away 3"? The set of whole numbers needed to be expanded to include the answers to such questions. Thus, the set of integers were devised to fill this gap and were defined as the set { . . ., -3, -2, -1, 0, 1, 2, 3 . . . }.

Adding and Subtracting Integers:
This is the point where some people lose touch with math. Some become tentative when dealing with operations like adding and subtracting negative integers. Because the operation of adding or subtracting negative integers is an integral part of card counting, and specifically applies when keeping a running count, we will talk about four of the more confusing operations in greater detail:

Adding two negative integers:
A simple rule is to first add the number parts and then put a minus sign in front of the result.

For example, if you are using the High-Low and have a running count of -4. If two tens now come out the deck, we need to add the -2 for the two newly seen tens to the running count of -4. To do this, we could temporarily ignore the minus signs and simply add 2 + 4 = 6 and then afterwards put a minus sign back in front of the 6 to give us -6.

This method may be confusing to some people, so if one prefers they can take -4 + (-2) = -6. In either method, the end result is the same.

Adding a positive integer and a negative integer:
One method is to ignore the signs to begin with, find the positive difference between the number parts and then attach the sign of larger original number.

For example, if you are using the High-Low and have a running count of -6. If a six, five, three and two now come out the deck, we need to add the +4 of the newly displayed cards to the -6 running count. To do this, we temporarily ignore the sign and find the positive difference between 6 and 4 which turns out to be 2. We then attact the sign of the larger original number. The larger original number was -6, so because it is a negative value, we attach the minus sign to our result of 2 and end up with -2.

Subtracting a negative integer from a negative integer:
The method is to first turn the subtraction into addition and then treat the question as described in the "Adding a positive integer and a negative integer" section.

Let's find the answer to the following; -5 - (-4) = ?

Subtracting a negative is the same thing as adding a positive, so we can rewrite -5 - (-4) = ? as -5 + 4 = ?
Now we just have to follow the rules for adding a positive integer to a negative integer. We find the positive difference between 5 and 4 is 1, and the negative 5 outweighs the positive 4, so the answer is negative. Therefore, we can say that:

-5 -(-4) = -5 + 4 = -1.

Subtracting a negative integer from a positive integer:
This is not as complicated as it sounds. The method used is to first turn the subtraction into an addition and then simply add the two values.

For example, if we want to find the answer to the following: +7 - (-4) = ?

We start by rewriting the equation as follows: +7 - (-4) = +7 + 4 = ?

Now we are faced with adding two positive integers, and 7 + 4 = 11.

Multiplying integers:
So, we have discovered that the set of integers is capable of handling the operations of addition and subtraction. How about multiplication? There are two multiplication operations that we will talk about:

Multiplying a negative integer and a positive integer:
I will assume that you know how to multiply two positive integers. When multiplying a negative integer with a positive integer, the product is negative.

For example, 4 * (-3) = -12.

Multiplying two negative integers:
The rule for muliplying two negative integers together is to treat the product as a positve and to multiply the two integers together without regard for signs.

For example, (-2) * (-3) = +6.

Multiplying a string of positive and negative integers:
The method of multiplying signed numbers is to attach a minus sign to the product if there is an odd number of negatives, and to treat the product as positive if there is an even number of negatives.

For example, (-4) * (-2) * (-6) * 2 = ?

First multiply the number parts, 4 * 2 * 6 * 2 = 96

Then go back and notice that there are three negatives. Because that is an odd number of negatives, the product is negative. Therefore, the answer is -96.

Dividing integers:
So, we have discovered that the set of integers is capable of handling the operations of addition, subtraction and multiplication. The rules for dividing are similar to the rules for multiplication. When dividing signed numbers, the result is negative if the original numbers have different signs, and the result is positive if the original numbers have the same sign.

For example, (-9) / (-3) = ?

First divide the number parts: 9 / 3 = 3. Then go back and look at the signs of the original numbers. In ths example, they both have minus signs so the result is positive: (-9) / (-3) = +3

Rational Numbers:
Sometimes the division of integers does not always result in an integer value. If we pick a couple of integers at random, say 12 and 5, we find that dividing 12 by 5 produces a non-integer result. Mathematicians have called the ratio of two integers a "rational number." All integers fall into this category because they can be written as ratios of themselves to 1. For example, 7 is equivalent to 7/1. Likewise, all fractions, all finite decimals, and all repeating decimals can be written as ratios of integers. Therefore, "rational numbers" accomodate the results of addition, subtraction, multiplication, and - with one exception - division of integer values.

Division by zero:
You may recall that someone somewhere has stated that division by zero is undefined. Well, Can't someone just define it? Wouldn't it be logical enough for someone to just say that 24 divided by 0 is 0?

The answer is "No." When we ask, "What is 4 divided by 2?" we are actually asking, "What number times 2 equals 4?" The answer is 2, because 2 times 2 equals 4. So when we ask "What is 24 divided by 0?" we are essentially asking, "What number times 0 equals 24?" Obviously there is no such number.

But what about 0 divided by 0? Can this case be defined? Well, when we ask, "What is 0 divided by 0?" we are asking, "What number times 0 equals 0. The answer could be any number because any number times 0 equals zero. Let's say one arbitraily defined 0 / 0 to be 0. There is a rule that says that 0 divided by anything equals zero. So far, so good. But there is another rule that says anything divided by itself is 1. So, there is a rule conflict and because there is no way to decide between these rules, we say that 0 divided by 0 is undefined.

Irrational Numbers:
The set of rational numbers covers all kinds of operations. It covers addition, subtraction, multiplication and division. Just when you may have been thinking that the set of rational numbers could contain the results of all mathematical operations along comes the "square root."

If we pick an integer that happens to be a perfect square, then taking the square root of the integer will result in another integer. For example sqrt(9) = 3, or sqrt(25) = 5.

But if we pick an integer that does not happen to be a perfect square, then we end up with a non-repeating decimal. AND non-repeating decimals cannot be written precisely as a ratio of integers. For example, the sqrt(2) is only approximately equal to 1.4142135624. These kinds of numbers (ie numbers that cannot be written precisely in fraction or decimal form) are called "irrational numbers."

Real Numbers:
Obviously, the set of rational numbers had to be expanded to include all numbers with non-repeating decimals. Yet square roots are not the only irrational numbers. Cube roots, fourth roots, and higher roots also produce irrational numbers. Moreover, two of the most well-known constants in mathematics "pi" and "e", are irrational as well. Mathematicians decided to call the expanded set of numbers, which includes both rational and irrational numbers, the set of Real Numbers."

Imaginary and Complex Numbers:
Here we go again. Even the set of real numbers was not big enough to include taking the square root of negative numbers. Let's take sqrt(-9) as an example. There is no real number you can multiply by itself to produce -9. Therefore, to accomodate the square roots of negative numbers, mathematicians came up with the concept of imaginary numbers and complex numbers.

Imaginary numbers are of the form a*i, where "a" is a "real number" and "i" is the square root of -1.

Complex numbers are of the form a + b*i, where "a" and "b" are real numbers and "i" is the square root of -1.

For now, don't worry about imaginary or complex numbers. I have included the information just in case one has heard about them and was curious about where they fit into the overall number system.

Summary:
The study of numbers provides the basis upon which to build. Numbers are to mathematics like letters are to "English." Operations such as addition and subtraction are like combining letters into words. Equations are like sentences. Theorems are like paragraphs. Proofs are like essays.

Mathematics is a wonderfully complex subject. The starting point is the subject of numbers and hopefully, this article has provided a glimpse into the rationale and the terminology necessary to understand the numbers we use in the mathematics of Blackjack.