 Our first use of mathematics, and how far away is it, was the direct measurement of the distance between where I was standing and a pillar in my backyard. The mathematical foundations for this go to the number line, which is based on our concept of numbers, so we'll start with the number system. In the beginning, across all ancient cultures, there were natural numbers, or counting numbers. Like eggs in a basket, we can have one, or one more than that would be two, or one more than that would be three, etc. The beauty of mathematics is that it is so versatile. It can not only be used to count eggs, it can count money, or stars in our galaxy, or even galaxies within a range of redshifts to determine whether or not the universe is flat. The counting numbers did not include the number zero. You cannot count zero. Its origins date back to a famous ancient Indian scroll called the Bakshali manuscript, created 1,600 to 1,700 years ago. Back then it was written as a dot used as a placeholder for numbers larger than nine. If we add the number zero to our set of counting numbers, we get the set of whole numbers. There are multiple sets of symbols for these numbers. The most famous are the Roman numerals and the ten repeating positional Hindu-Arabic numerals. I've included the two repeating positional computer digits to highlight different base counting. The Hindu-Arabic numerals are by far the most useful and replace Roman numerals when the Roman Empire fell around 300 CE. The Hindu-Arabic number system has some interesting properties. Here's one of them. Multiply nine times any number, say 983,264. Pick one of the digits in the product. We'll pick the number seven. Now add up the remaining digits. If the answer has more than one digit, add them. Repeat the process until you get down to a single digit. Subtract that digit from nine, and you get the number you've picked out of the original product. This will work no matter which digit you choose to remove from the product. It's the base ten system that makes this work. For example, we can write the number ABCDEF as the sum of its positional digits. With this view, we can see that the numbers two, five, and ten divide evenly into each of the terms except possibly the last digit, F. This is why, if the last digit is even, the whole number is even. If the last digit is odd, the whole number is odd. If the last digit is divisible by five, then the whole number is divisible by five. And if the last digit is zero, the whole number is divisible by ten. Moving on a bit, we can rewrite this number by breaking up the powers of ten, subtracting and adding one to each of them. Multiplying each digit through the sum and rearranging, we get this. Every number in the first bracket is divisible by nine. So if the sum of the digits in the second bracket is also divisible by nine, the whole number is divisible by nine. Furthermore, if the sum in the second bracket is not divisible by nine, then the whole number is not divisible by nine. By the way, the same thing is true for three. So in our missing digit exercise, we multiplied a number by nine, guaranteeing that the sum of the product's digits would add up to nine. Now pulling out any digit will reduce the remaining sum by just that amount. So subtracting it from nine gives you the removed digit. This system is easily extended into the decimal number system by dividing by ten for each position to the right of the decimal point in much the same way we multiplied by ten for positions to the left. We write abc.efg as this. For example, 0.75 is 7 over 10 plus 5 over 100 or three fourths. If we include negative numbers along with positive numbers and zero, we get the full set of integers, but it took a long time to fully accept the concept of negative numbers. The ancient Greeks did not have them. The earliest written reference to negative numbers was found in the Chinese book The Nine Chapters on Mathematical Art, written around 100 BCE. In fact, negative numbers were not fully accepted until the 19th century. After all, you can't have less than zero eggs in the basket. Here's an illustration that highlights the problem mathematicians had with negative numbers. We use this in the How Fast Is It? video book explaining the theory behind the Michelson-Morley experiment. We have a boat in a river traveling upstream with a motor that can propel it at a steady speed in still water. The river is flowing in the opposite direction. The boat's home base is a known distance away. The question is, how long will it take the boat to get home? The solution is pretty straightforward. The time it takes is just a distance it has to travel divided by the speed it is traveling. And that speed would be its velocity minus the velocity of the river. If the distance is 30 kilometers and the boat is running at 20 kilometers per hour and the current working against it is 5 kilometers per hour, we see that the trip home will take two hours. But what if the current is greater than the speed of the boat, say 25 kilometers per hour? Then the equation gives us negative time. Is time running backwards? Absurd. But if we apply the math to the situation we used to develop the equation, we see that a negative time simply means that the poor boat will never make it home. The river will simply continue to carry it downstream. The number associated with a point is called its coordinate. We then associate the number zero with the origin and then extend the line to the left in the same sized segments we counted off on the right. These points correspond to the negative numbers where the leftmost point in the first segment represents minus one. The leftmost point in the second segment represents minus two and so forth. This is the basic number line for the set of all integers. In order to indicate that these numbers are carried out to the right and to the left without limit, we introduce the symbols plus infinity and minus infinity. But we need to keep in mind that these are not numbers. With the set of integers in hand, we can define the four basic arithmetic operations of addition, subtraction, multiplication, and division. We can define addition as an exercise in counting. A plus B means start with the number specified by the first term, A, and count the number specified by the second term, B. In this example, we start with A at eleven and count seven more, or B. Zero is the additive identity in that adding zero to any number gives you the same number you started with. Subtraction would then be start with the first term and count backwards the number specified by the second term. Here we are counting seven times back from eleven. Thus addition and subtraction are tied directly to counting, and counting has been shown to accurately represent anything you can count. We can also define multiplication in terms of addition, which is based on counting. A times B says add the number A to itself, B times. Here we are starting with three and adding it to itself five times. Doing it one time leaves it unchanged. Thus the number one is the multiplicative identity, like zero was the additive identity. But what does it mean to add a number to itself zero times? To deal with this, we define a number being added to itself zero times to be the number zero. We define division as the inverse of multiplication. For the division of one number, say A, by another number, say B, we are asking how many times can we subtract B from A? Or more generally, what number, when multiplied by B, gives us A? For example, we would ask how many times can we subtract three from fifteen, and the answer is five. In the late 1800s, the mathematician Giuseppe Piano produced a set of axioms or postulates that can be used to develop number properties. In simple terms, starting with the counting numbers they are as follows. One is a number. Every number N has one and only one successor number, N plus one. And no two different numbers have the same successor number. In other words, if N plus one equals M plus one, then N has to equal M. From these postulates and our basic operator definitions, a number of properties exist that we can use to manipulate numbers and solve equations. Here are the properties for the natural or counting numbers. Although we tend to take them for granted, mathematicians have to prove each and every one. You'll find a proof that A plus B equals B plus A for natural numbers in the appendix. Natural numbers are closed for addition and multiplication. By closed, we mean that these operations on numbers in the set produce numbers that are also in the set. When we include zero and negative numbers, we get the integer number line. The set of integers adds closure for subtraction. The set of integers is not closed with respect to division. One divided by two is not in the set of integers. To include these and make the set close with respect to division, we need to add all the rational numbers. The numbers that can be expressed as a ratio of integers or the denominator is not zero. To map these numbers to the number line, we simply divide each segment into the number of sub-segments indicated by the denominator. For example, here's nine fourths. The line that contains all integers and rational numbers and zero is known as the rational number line. It has all the properties of the integer number line and is closed for all the basic operations. It not only contains all rational numbers, it provides for ordering. One number is greater than another if its coordinate on the line is to the right. It is less than another if its coordinate is to the left and it is equal to the other if its coordinate is at the same location. With the rational number line in hand, we can now define distance on the line. If p and q are rational points on the line with coordinates x and y respectively, such that x is less than or equal to y, then the distance between p and q is y minus x. It is straightforward to find the midpoint between any two rational points p and q. We'll call it m with a coordinate equal to t. The distance to t would be the distance to p plus half the distance between p and q. That would be y minus x divided by 2. We see that the coordinate of the midpoint of two points on the number line is half the sum of the given points. Consequently, there is always a rational point between any two rational points on the line. Because we can divide by two without limit, it follows that there are infinitely many rational numbers between any two given rational numbers no matter how close together they are. We say the rational number line is dense. An important consequence of the set of rational numbers being dense is that the length of any segment can be approximated to any degree of accuracy by a rational number. And there is one more important point to make about this line. If we shift the origin, every coordinate on the line will change. For example, if we shift it to the right two units, the coordinates on the new line will be different from the originals by two units. But there is one thing that does not change when we change the coordinates. And that is the length of any line segment. It is said to be invariant with respect to coordinate transformations. You can see how the shift in the values of the coordinates cancel out in the length calculation. Before we leave the number line, there are two more considerations we need to examine. One, there are missing points on our rational number line. And two, the number zero has some unique and relevant characteristics. Epossas, born around 500 BCE, was a Greek philosopher of the Pythagorean School of Thought. He is widely regarded as the first person to recognize that a square's diagonal cannot be expressed as the ratio of two integers. At this time in Greek society, numbers were intimately connected to their religion. So Epossas's finding was considered heresy. 200 years later, Euclid published his proof. Here's how it goes. Let's assume that there is such a rational number and show a resulting contradiction that negates the assumption. Suppose p over q is a rational number expressed in its lowest terms, meaning they have no common factors except the number one, such that p over q is equal to the square root of two. We can square both sides of the equation and multiply both sides by q squared. This shows that p squared is an even number, and therefore p must be an even number because an odd number times itself would be an odd number. Since p is even, there exists a number t, such that p is equal to two times t. If we substitute this in for p and divide both sides by two, we see that q is also an even number. In other words, both p and q have two as a factor. But our stipulation was that they had no factors in common except for the number one. We have a contradiction, and it shows that the statement the square root of two can be expressed as a rational number is false. Therefore it cannot be expressed as a rational number. In fact the nth root of any number that isn't a perfect n square is irrational. Add to that the fact that any irrational times irrational will be irrational, and you can see that the set of irrational numbers is infinite. The rational number line is dense, but it is not continuous. The union of the set of rational numbers and irrational numbers creates the set of real numbers. But to prove the basic number properties for a number line that includes irrational numbers turned out to be quite the problem. In 1872 Richard Dedekind defined cuts in the rational number line that exposed the holes created by irrational numbers. He then proved that the set of these cuts is equivalent to the set of real numbers. This extended the rational number line into the real number line in a manner that preserved all the properties of the rational number line. In addition, it is not only dense, it is continuous, it has no holes. The real number line is the foundation from which all the rest of our math will flow. Here's a basic algebraic exercise that illustrates the issue covered in the next segment. We can multiply both sides of the equation by A. We can subtract B squared from both sides. We can factor A squared minus B squared into A plus B times A minus B on the left hand side of the equation, and A B minus B squared into B times A minus B on the right side of the equation. We can divide both sides by A minus B, and we can cancel out common terms in the fractions. So the equation simplifies to A plus B equals B. Substituting in the ones for A and B, we get 1 plus 1 or 2 equals 1. This happened because we divided both sides of the equation by A minus B, which equals 0. In this way, dividing by 0 is like a box of chocolates. You never know what you're going to get. A closer look at the number 0 will explain why. It's important to understand that division has a problem when it comes to the number 0. Suppose A is a number not equal to 0. Then, for A divided by 0, we're asking what number is when multiplied by 0 will give us A. But no matter what number we multiply by 0, you will always get 0, never A. So A over 0 has no meaning. We say it is undefined. Now if A is 0, we're asking what number, when multiplied by 0, would give us 0. The answer is any number at all. Because any number multiplied by 0 would give us 0. This makes 0 divided by 0 completely undetermined. It can be any number you can think of. This is what gave us the 1 equals 2 result. The problem with 0 also shows up with exponents. For any number A and any positive integer n, we define A, the base, raised to the nth power, the exponent, to be the base multiplied by itself the number of times specified by the exponent. We see that when we multiply two numbers with the same base, we can add the exponents. With that in mind, we define a negative exponent to mean 1 divided by the base raised to the positive value of the exponent. This extends the exponent arithmetic to include all integers. It follows that A raised to the nth power times A raised to the minus nth power will equal the number 1. It also follows that A raised to the nth power times A raised to the minus nth power will equal A raised to the power of 0. So in order for the arithmetic to hold, we must define a number multiplied by itself 0 times to equal the number 1, the multiplicative identity. This is much like adding a number to itself 0 times is equal to 0, the additive identity. But we also know that if the base is 0, raising it to any power will always give you 0. So what if both the base and the exponent are 0? Does 0 raise to the 0 power equal 1? Or does 0 raise to the 0 power equal 0? It is said to be indeterminate. So when we apply math to a physical situation, we must always take care to never wind up dividing by 0. We must always stipulate the ranges where an equation is operative and where it is not. We can now take a look at the measurement we started with. If we mark the equal segments on the measuring tape to be meters, we get 2 meters. If we mark the equal segments to be feet, we get 6 and 56 one hundredths feet. Here we need to highlight a key difference between the pure mathematics and the physics of measuring distances. Math uses exact coordinates to give us the exact distances. But physical measurements always involve a level of inaccuracy. For completeness purposes, you will often see a distance accompanied by an estimated error magnitude. In this case we would say that the distance to the pillar is 2 meters plus or minus 1 millimeter. Or 2.19 yards plus or minus some fraction of an inch. Depending on the accuracy of the tape measure. Here's a way to do multiplication using doubling, having and adding. To illustrate let's multiply 127 by 46 the old way. First we multiply by 6 and then by 40. We add the two together for the result. In this method we need two columns. Now pick one of the two numbers to be multiplied. We'll use the 46. Put it at the top of the first column. Divide it in half. If it were an odd number, we'd subtract 1 to get an even number and then divide that in half. Continue this process until you get to the number 1. Now place the other number, 127 in our example, at the top of the second column. Double it and double it again and again until you reach the row with the number 1 in the first column. Now scratch out each row that has an even number in the first column and add the remaining numbers in the second column. This is the product. It is quite common in mathematics to use inductive reasoning for proofs of the sort where you're trying to prove something is true for all numbers in an infinite set. Proving the commutivity property for addition A plus B equals B plus A for all counting numbers is one of them. All we have to go on are our postulates and one proven theorem. The postulates are 1 is a number. Every number n has 1 and only one successor number n plus 1. No two different numbers have the same successor number. And the associative property of addition A plus B plus C is equal to A plus B plus C. The commutative property proof has two parts. In the first part we show that A plus B equals B plus A for any value of A when B equals 1. Our inductive assumption is that for some value of A the relationship is true. We'll show that that implies that the relationship is then true for A plus 1. We want to show that A plus 1 plus 1 is equal to 1 plus A plus 1. We start with A plus 1 plus 1. We can set it equal to 1 plus A plus 1 by our inductive assumption. And we can change that to 1 plus A plus 1 by the associative property and we're done. Now we'll show that A plus 1 equals 1 plus A when A equals 1. We start with A plus 1, substitute in 1 for A and then substitute in A for the other number 1 and we're done. So it is true for A equals 1 and by the first result we know that it is true for A equals 1 plus 1, that's 2 and then for 2 plus 1, that's 3, etc. for all counting numbers. Now in part 2 our inductive assumption is that A plus B equals B plus A for some value of B. We'll show that this implies that A plus B plus 1 is equal to B plus 1 plus A. So we'll start with A plus B plus 1. We can use the associative property around A and B. We can then use the inductive assumption to get B plus A plus 1. Reuse the associative property to get B plus A plus 1 and then use the results of part 1 that showed A plus 1 was equal to 1 plus A to get B plus 1 plus A and then use the associative property again to get B plus 1 plus A. And we're done.