1. The Dawn of Numbers
A humorous look at early attempts at creating number systems, leading up to our modern base-10 decimal number system which uses 'positional notation'. The story takes place on the fictitious island of Cocoloco.
2. Roman Numerals: Sign-Value versus Positional Notation
Roman numerals are an ancient base-10 natural number system. Understanding Roman numerals (a sign-value notation) can shed light on our modern number system which uses positional notation.
3. Decimal, Binary, Octal & Hexadecimal
Our modern decimal number system is base-10. Other number systems used in fields like computer engineering are base-2 (binary), base-8 (octal) and base-16 (hexadecimal).
4. Whole Numbers, Integers, and the Number Line
Number systems evolved from the natural 'counting' numbers, to whole numbers (with the addition of zero), to integers (with the addition of negative numbers), and beyond. These number systems are easily understood using the number line.
5. Commutative & Associative Properties of Addition
A look behind the fundamental properties of the most basic arithmetic operation, addition.
6. Commutative Property of Multiplication
The commutative property is common to the operations of both addition and multiplication and is an important property of many mathematical systems.
7. Associative and Distributive Properties of Multiplication
A look at the logic behind the associative and distributive properties of multiplication.
8. Multiplying Negative Numbers
When number systems were expanded to include negative numbers, rules had to be formulated so that multiplication would be consistent regardless of the sign of the operands.
9. Division and Prime Numbers
The building blocks of all natural numbers are the prime numbers. The early Greeks invented the system still used today for separating natural numbers into prime and composite numbers.
Any natural number can be decomposed into a product of prime factors. Prime factorization is fundamental to many arithmetic operations involving fractions.
11. Fractions and Rational Numbers
The first fractions used by ancient civilizations were 'unit fractions'. Later, numerators other than one were added, creating "vulgar fractions" which became our modern fractions. Together, fractions and integers form the "rational numbers".
12. Arithmetic Operations with Fractions
Arithmetic operations with fractions can be visualized using the number line. This chapter starts by adding fractions with the same denominators and explains the logic behind multiplication of fractions.
13. Reciprocals and Division with Fractions
When working with fractions, division can be converted to multiplication by the divisor's reciprocal. This chapter explains why.
14. Creating Common Denominators
Addition and subtraction of fractions with different denominators requires creating a 'common' denominator. Using the number line, this mysterious process can be easily visualized.
15. Creating Least Common Denominators
Sometimes when finding a common denominator we create an unnecessarily large common denominator. This chapter explains how to find the smallest possible common denominator.
16. Reducing Fractions
The process of reducing any fraction to its simplest possible form is easily visualized using the number line.
17. Improper Fractions and Mixed Numbers
Sometimes arithmetic operations result in fractions greater than one, called 'improper' fractions. An improper fraction can be converted into a 'mixed number' composed of an integer plus a 'proper' fraction.
18. Converting Fractions to Decimal Numbers
Any fraction can be converted into an equivalent decimal number with a sequence of digits after the decimal point, which either repeats or terminates. The reason can be understood by close examination of the number line.
19. Converting Terminating Decimal Numbers to Fractions
Decimal numbers with a finite number of digits after the decimal point can be easily converted into fractions. This chapter explains why.
20. Converting Repeating Decimal Numbers to Fractions
Decimal numbers with an infinitely repeating sequence of digits after the decimal point can be converted into fractions. This chapter explains why.
Exponentiation is shorthand for repeated multiplication, just like multiplication is shorthand for repeated addition. Multiplied or divided exponential terms with like bases can be combined by adding or subtracting their exponents.
22. Exponents of One, Zero, and Negative
Integer exponents greater than one represent the number of copies of the base which are multiplied together. But what if the exponent is one, zero or negative? Using the rules of adding and subtracting exponents, we can see what the meaning must be.
23. Scientific Notation
Scientific notation allows us to more easily express very large or very small numbers encountered in engineering and science. Using exponents, we can convert standard decimal numbers into scientific notation and vice versa.
24. Simplifying Multiplied Exponential Expressions
Exponential expressions with multiplied terms can be simplified using the rules for adding exponents.
25. Simplifying Divided Exponential Expressions
Exponential expressions with divided terms can be simplified using the rules for subtracting exponents.
26. Simplifying Mixed Exponential Expressions
Exponential expressions with multiplied and divided terms can be simplified using the rules of adding and subtracting exponents.
27. Raising Exponential Expressions to Powers
If a term raised to a power is enclosed in parentheses and then raised to another power, this expression can be simplified using the rules of multiplying exponents.
28. Raising Products and Quotients to Powers
Any expression consisting of multiplied and divided terms can be enclosed in parentheses and raised to a power. This can then be simplified using the rules for multiplying exponents.
29. Roots and Unit Fraction Exponents
Exponents can not only be integers. They can also be fractions. Using the rules of exponents, we can see why a number raised to the power 'one over n' is equivalent to the nth root of that number.
30. Rational Exponents
Exponents can not only be integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.
31. Simplifying Radical Expressions
Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.
32. Irrational Numbers
Although the Greeks initially thought all numeric qualtities could be represented by the ratio of two integers, i.e. rational numbers, we now know that not all numbers are rational. How do we know this?
33. Real Numbers
There are an infinite number of rational numbers, but there are infinitely more irrational numbers. Neither type of number can represent every type of numeric quantity. By combining the rational and irrational numbers into the real numbers, a continuum of numbers is created which can represent any quantity in the real world.