Chapter 0: Introduction to Numbers and Arithmetic

0.1 Arithmetic with integers

  • Origins of number notation
    • Numbers began as counting numbers or positive integers.
    • Romans had counting numbers but no zero; used repetition of symbols for larger numbers and added new symbols when the list grew.
    • Examples of Roman numerals: I, II, III, IV, V, X; IX is nine; XX is twenty.
    • Additional symbols: L for fifty, C for one hundred, D for five hundred, and M for one thousand.
    • The concept of zero came to Western mathematics from the Arabs (who likely got it from Hindu mathematicians).
    • Zero enables numbers like 10, 100, 1000, 10^4, … and allows consistent rules for arithmetic, including negative numbers.
    • Key implication: zero makes possible a full number system that supports addition, subtraction, multiplication, and later division.
  • Addition, subtraction, and negative numbers
    • With counting numbers, addition is straightforward: e.g., counting 2 items then 3 items gives 5.
    • Memorizing the addition table (50+ pairs) is a starting step; there are 55 pairs in total, but many involve 0 which are simple.
    • Definition of 0: the number that, when added to anything, yields no change.
    • Closure: A number system is closed under an operation if the result of the operation is always a number within the system.
    • Counting numbers alone are not closed under subtraction (e.g., starting with 3 and subtracting 5 yields a non-counting-number).
    • Negative numbers: interpreted as debts or losses; positive numbers as gains.
    • Examples:
    • 8 + (-5) = 3. (EQ0.1)
    • Subtracting a negative: -(-5) = 5, so 8 + (-(-5)) = 8 + 5 = 13. (EQ0.2)
    • Subtraction is the inverse of addition; this inverse relationship is summarized algebraically in Appendix 1 of the book.
    • Integers: the set consisting of counting numbers, their negatives, and zero; provides closure for both addition and subtraction.
  • Multiplication
    • Symbols for multiplication include x, *, and the dot ·; sometimes the symbol is omitted in algebra (e.g., vt meaning v multiplied by t).
    • Caution: computer programs require explicit multiplication operators (e.g., in spreadsheets, (a + b)(c + d) must be written as (a + b) * (c + d)).
    • Multiplication is repeated addition: 5 × 2 means adding 2, five times, giving 10; equally 2 × 5 equals 10.
    • You can verify this by counting blocks or imagining a rectangle with 2 blocks across and 5 blocks down (see Figure 0.3).
    • In principle, multiplication is commutative: 5 × 2 = 2 × 5.
    • The number 0 is special: in multiplication, it annihilates numbers (0 × a = 0). The multiplicative identity is 1 (1 × a = a).
    • The identities that formalize these properties are given in Appendix 1.
    • Combining addition and multiplication: the distributive property.
    • Example: add five 2s and five 3s:
    • 5 × 2 + 5 × 3 = 5 × (2 + 3) = 5 × 5
    • Another example: 27 + 18 = 3 × 9 + 2 × 9 = (3 + 2) × 9 = 5 × 9 = 45
    • The idea of factorization: breaking a number into factors and regrouping terms to simplify expressions.
  • The multiplication triangle (Figure 0.4)
    • Describes that the number at the intersection of a row and a column equals the product of the numbers at the top of the column and the left of the row.
    • Aids in rearranging algebraic statements to gain insights; the text notes we will return to this idea later.
  • Rules for multiplication with negative numbers and their relation to addition/subtraction
    • The rules for multiplying negatives follow from the rules of addition and subtraction.
    • Examples:
    • (-5) × 2 = -10 (subtracting 2, five times).
    • (-5) × (-2) = +10 (removing debits corresponds to adding credits).
    • This demonstrates that removing a negative is equivalent to adding a positive.
    • These rules are consistent with the idea that multiplication distributes over addition and respects sign changes.

0.2 Division, reciprocals, fractions, and rational numbers

  • The integers are closed under addition, subtraction, and multiplication but not under division.
  • Division concept
    • Division asks: how many b’s are in a? If a = b × c, then a ÷ b = c.
    • In general, division and multiplication must reverse each other.
    • Examples:
    • 10 ÷ 2 = 5 and 10 ÷ 5 = 2 (these are exact in integers).
    • 10 ÷ 3 is not an integer (there are more than three but fewer than four 3’s in 10).
    • 2 ÷ 9 is less than 1 but greater than 0.
  • Reciprocals and fractions
    • To handle divisions that do not yield an integer, we extend to reciprocals and fractions.
    • The reciprocal of an integer n is 1/n, defined so that n × (1/n) = 1 (provided n ≠ 0).
    • Example: 7 × (1/7) = 1.
    • A fraction is any integer multiple of a reciprocal. For example, 2 × (1/7) = 2/7, 3 × (1/7) = 3/7, etc.
    • The numerator is the number above the line in the fraction; the denominator is below.
    • 2 × (1/7) of a whole pie is the same amount as (1/7) × 2 of two whole pies, i.e., 2/7 of a pie.
  • Division reinterpreted via fractions
    • Example: 3 ÷ 2. If you have 3 pies and cut each into 2 equal pieces, you have 6 pieces; dividing into two piles yields 3 pieces per pile, i.e., 3 × (1/2) = 3/2.
  • Important exception: division by zero is not allowed.
    • The reciprocal of 0 does not exist, so division by 0 is undefined.
  • The order of division and reciprocals
    • 2 ÷ 3 = 2 × (1/3) = (1/3) × 2 = 2/3.
    • 3 ÷ 2 = 3 × (1/2) = (1/2) × 3 = 3/2.
    • Note: when writing a ÷ b, you use the reciprocal of b; the result depends on which number is in the denominator.
  • Ratios and rational numbers
    • Chapter 2 will define a ratio as something like 3:4, read as 'three to four'.
    • Mathematicians view a ratio as a single number divided by another, so 3:4 = 3/4.
    • A fraction is an integer times the reciprocal of an integer, and thus a ratio of integers.
    • Rational numbers are numbers that can be expressed exactly as p/q with p, q integers (q ≠ 0). This includes all integers, since p/1 = p.

0.3 Decimal numbers

  • Decimals are a compact way of writing fractions with denominators that are powers of 10, e.g., 10, 100, 1000.
    • Example: 173.8 = 1738/10; 17.38 = 1738/100.
  • Once you can do arithmetic with fractions, you already have the rules for decimals.
  • Rational numbers have decimal representations that are either exact decimals or repeating decimals.
  • Examples of exact decimals and repeating decimals
    • 1/2 = 0.5 exactly
    • 1/8 = 0.125 exactly
    • 1/125 = 0.008 exactly
    • 1/3 = 0.333333… (3 repeats forever): rac{1}{3} = 0.
      \overline{3}
    • 1/7 = 0.142857142857… (142857 repeats).
      17=0.142857‾\frac{1}{7} = 0.\overline{142857}
    • 9/14 = 0.642857142857… (the digits 642857 then 142857 repeat, pattern is the repeating block 142857 after the initial 0.6):
      914=0.642857142857…\frac{9}{14} = 0.642857142857…
  • Rational vs. irrational decimals
    • Rational numbers have decimal expansions that either terminate or repeat.
    • Irrational numbers do not have repeating decimals and cannot be written exactly as a fraction.
  • Examples of irrational numbers (to come later)
      • -
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  • Implication for science
    • Real numbers (rational + irrational) describe physical quantities to any desired precision; measurements are approximations to underlying real values with units.

0.4 Real numbers

  • Real numbers consist of all rational and irrational numbers.
  • Decimal representations that repeat forever are still rational because they correspond to fractions; irrational numbers cannot be represented exactly as fractions.
  • Examples of irrational numbers mentioned later include:
    • (\sqrt{2}) (sqrt(2))
    • (\pi) (the ratio of a circle's circumference to its diameter)
    • (e) (the base of natural logarithms)
  • Rational numbers are central to practical arithmetic (marks on paper, calculators, and computers).
  • In science, real numbers are used to describe quantities; however, irrational numbers require approximations in practice.
  • Approximations and representations
    • (\sqrt{2}) cannot be written exactly as a fraction; it must be approximated by rational numbers.
    • It is common to write something like (\sqrt{2} \approx 3.1416) in the wrong example; the correct context is that (\pi \approx 3.1416).
    • When approximations are used, notation such as (\sqrt{2} \approx 1.41421356\dots) may be employed depending on required accuracy.
  • Real numbers and measurement
    • Real numbers provide the theoretical basis; measurable quantities are expressed as real numbers with units, as discussed in Chapter 1.
  • Note on exactness and symbolic use
    • In some contexts, a symbol like (\pi) is used to denote the irrational value when exact equality is not needed; in other contexts, decimals are used for numerical computations.

0.5 Forbidden operations using real numbers: complex numbers

  • Two operations that cannot be performed with just real numbers
    • Division by zero: for any real a and b, a ÷ b is defined to satisfy a = b × c, but if b = 0 there is no unique c; division by zero is undefined.
    • The square root of a negative number: (\sqrt{-1}) has no real value.
  • Limits and division by zero
    • When both a and b approach zero (0/0), the limit depends on the path and requires more advanced analysis; this was a fundamental issue addressed by Leibniz and Newton via differential calculus.
  • Complex numbers
    • To handle square roots of negative numbers, extend the number system to complex numbers.
    • A complex number has two components: a real part and an imaginary part, written as (a + bi) with (i^2 = -1).
    • All arithmetic operations (except division by zero) on complex numbers are closed within the complex numbers.
  • Why complex numbers? Applications and caveats
    • Physicists: simplify calculations in optics, electrical circuit theory, and quantum mechanics.
    • Biologists: occasionally used in signal analysis and spectroscopy techniques like NMR.
    • For many readers of this book, real numbers suffice, so the book focuses on real numbers.

0.6 Summary

  • Classifications of numbers touched on in this chapter
    • Counting numbers (positive integers)
    • Integers: positive integers, zero, and negative integers
    • Fractions: numbers of the form (p/q) with integers (p, q) and (q \neq 0); includes all integers since (p/1 = p)
    • Rational numbers: numbers that can be written as a fraction and include all numbers expressible as (p/q)
    • Irrational numbers: cannot be written as a fraction; examples include (\sqrt{2}), (\pi), and (e)
    • Real numbers: union of rational and irrational numbers; describe physical quantities together with units
  • Real start of the book
    • The chapter sets the foundation for describing physical quantities and units, leading into the practical applications covered in the next chapter.
  • Additional context
    • The chapter references historical figures (e.g., Newton) and emphasizes the evolution of number systems from counting numbers to the integers, rationals, reals, and complexes, along with the practical implications for science and computation.