This book is designed to answer questions like the following. They are questions that are usually stated without proofs in elementary presentations. Note that there may be more than one correct answer. This book gives a single coherent development of the properties of numbers such that the answers to these questions are given by (almost) as few basic assumptions and derivations as possible.
The natural numbers are symbols from the sequence 1, 2, 3, … etc. Addition and multiplication are operations that replace two natural numbers with a single number. Addition is denoted by "+", and multiplication is denoted by "×" or "·". Sometimes the multiplication sign is omitted, as in y = a x.
How can addition and multiplication of natural numbers actually be defined for an infinite (unlimited) set of numbers?
Why is addition commutative and associative; i.e., why is x + y = y + x, and x + (y + z) = (x + y) + z?
Why is multiplication commutative and associative; i.e., why is x × y = y × x, and x × (y × z) = (x × y) × z?
Why does multiplication distribute over addition; i.e., why does x × (y + z) = (x × y) + (x × z)?
Why is 2 + 2 = 4? Clearly, this is a matter of definition, but definitions of what?
If x, y and z are natural numbers and x + z = y + z, then x = y. Why? This does not depend on one being able to substract z from each side of the equation.
Why is there only one way to factor a natural number?
The integers are built by adding an additive inverse -1 of 1 to the natural numbers, and then also including any number that can be written as the product or sum of -1 and a natural number. For example, 1 + (-1) defines an element that must be included, since it is the sum of -1 and a natural number. We define 1 + (-1) to be an element called 0 (zero), and we define x + 0 = x and 0 · x = 0. We define the additive inverse of x to be -x = (-1) · x. We define subtraction by x - y = x + (-y).
It is asserted that such a system can be defined such that addition and multiplication are communtative and associative, and that multiplication distribute over addition. Since these are properties of the natural numbers by themselves, it is asserted that the integers are an extension of the natural numbers; i.e., that the natural numbers are a subset of the integers. How do you know that this is possible?
Why is the product of two negative numbers positive?
The rational numbers or fractions are built from the integers by adding to each integer x ≠ 0 a number 1/x (the multiplicative inverse) such that x · (1/x) = 1. Then division is defined by x ÷ y = x · (1/y). Usually one sees the notation reduced even further to x ÷ y = x/y.
It is asserted that addition and multiplication can be defined for this expanded set of numbers such that these operations are commutative and associative, and that multiplication distributes over addition. How do you know that this is possible?
The integers are said to be well-ordered, but fractions are just ordered. What's the difference?
Why is (a ÷ b) ÷ (c ÷ d) = (a · d) ÷ (b · c)?
Why can't the square root of 2 be represented as a fraction?
The real numbers are an extension of the rational numbers so that the real numbers include the the rational numbers, plus numbers like the square root of 2. The real numbers that are not rational are called irrational numbers.
What is the defining property of the real numbers that sets it apart from the rational numbers?
How do you know that addition and multiplication for real numbers can be defined so that they are commutative and associative, and multiplication distributes over addition?
Why is e defined to be lim n→∞ (1 + 1/n)n, where n is a natural number?
How do you show that lim n→∞ (1 + 1/n)n actually defines a real number?
If em, where m is an integer, means that e is to be multiplied by itself m times, show that em = lim n→∞ (1 + m/n)n.
For a general real number x, ex = lim n→∞ (1 + x/n)n. The function exp(x) is defined to be 1 + x + (x2/2) + … + (xn/n!) + …, where n! = n · (n-1) · (n-2) · … · 1. Show that ex = exp(x).
The complex or imaginary numbers are built from the real numbers by adding the element i, defined by the property i2 = -1, along with any number which is the product or sum of i with a real number.
It is asserted that addition and multiplication can be defined for this expanded set of numbers such that these operations are commutative and associative, and that multiplication distributes over addition. How do you know if this is possible?
A complex number z can be represented as x + i y, where i y is understood to mean i×y, and x and y are real numbers. What are the additive inverses and multiplicative inverses of a complex number?
Show that any complex number can be represented as z = r · (cos(θ) + i sin(θ)), where cos(θ) = x/r, sin(θ) = y/r, and r2 = x2 + y2.
If en, where n is an integer, means that e is to be multiplied by itself n times, what is ei n? The point is that ez can only make sense by using the definition ez = lim n→∞ (1 + z/n)n. The notation ez is misleading in this case.
For any real numbers θ, how can you show that ei θ = cos(θ) + i sin(θ), where sin(θ) and cos(θ) are the usual trigonometric functions? Can you show that θ must be measured in radians for this to be true? The arguments in this book do not use differential or integral calculus to show this.
Can you show that (ez1) · (ez2) = e(z1 + z2)?
Can you show how the requirement that ln(z1 · z2) = ln(z1) + ln(z2) leads to the definition ln(z) = limn →∞n · (z1/n-1)?
Show that one can choose ln(z) = ln(r) + i θ, if one writes z = r · ei θ.
Can you show eln(z) = z and ln(ez) = z?
How do you calculate e and π just using addition and multiplication, and without using calculus, a calculator or looking them up in a book? This might take a lot of time, but that's OK. Remember, there were trig and log tables long before there were calculators. How could they have done this hundreds of years ago?
If z1 and z2 are complex numbers, what is z1z2? For example, what is the numerical value of ii?