Introduction to real numbers: from Sumerian tokens to R = Q ∪ Q′

Numbers took thousands of years

Your book traces numbers from ancient civilisations to the Arabic system we use today. Each people solved one counting problem and passed it on.

The Sumerians counted in base 60 with tokens: cones, beads and spheres for 1, 10, 60, 600, 3600 and 36000. The Egyptians used base 10 symbols, writing the highest place value first.

The Romans used seven letters, I, V, X, L, C, D and M, for 1, 5, 10, 50, 100, 500 and 1000. Indians gave zero and the decimal system; Arabs carried the numerals 0–9 to Europe.

Muhammad ibn Musa al-Khwarizmi made algebra a separate field in the 9th century. The counting set {1, 2, 3, …} was later extended to the real numbers: R = Q ∪ Q′, rationals together with irrationals.

Real numbers are the union of two sets: R = Q ∪ Q′. Every real number is either rational or irrational, never both.

Key terms

Sexagesimal

The Sumerian way of counting.

Rational number (Q)

A quotient of two integers.

Irrational number (Q′)

A number that cannot be written as a quotient of integers.

Real numbers (R)

All rationals and irrationals together.

Short questions with model answers

1. Q1. Write 2525 the Egyptian way your book describes (p. 1).

   • 2525 = 2000 + 500 + 20 + 5
   • 2000 → 500 → 20 → 5

   Symbols for 2000, then 500, then 20, then 5.

2. Q2. Using the Roman letters in your book, what is MDCLXVI?

   • M = 1000, D = 500, C = 100, L = 50, X = 10, V = 5, I = 1
   • 1000 + 500 + 100 + 50 + 10 + 5 + 1

   1666.

Common mistakes

• ✗ Saying the Arabs invented zero.

  ✓ Your book credits zero to the Indians; the Arabs carried the numerals to Europe.

• ✗ Saying the Sumerians used base 10.

  ✓ Sumerians used base 60; the Egyptians used base 10.

• ✗ Thinking a number can be rational and irrational at once.

  ✓ Q′ holds exactly the numbers that are not p/q, so the two sets never overlap.

MCQs

1. 1. The Sumerians counted in base:

   1. (a) 2
   2. (b) 10
   3. (c) 16
   4. (d) 60
   Show answer

   (d) Sexagesimal means base 60 (p. 1).

2. 2. Your book credits the concept of zero to the:

   1. (a) Romans
   2. (b) Indians
   3. (c) Egyptians
   4. (d) Sumerians
   Show answer

   (b) Indians developed zero and the decimal system (p. 2).

3. 3. The Roman letter D stands for:

   1. (a) 50
   2. (b) 100
   3. (c) 500
   4. (d) 1000
   Show answer

   (c) I, V, X, L, C, D, M = 1, 5, 10, 50, 100, 500, 1000 (p. 2).

4. 4. Who introduced algebra as a distinct field in the 9th century?

   1. (a) Al-Khwarizmi
   2. (b) Al-Karaji
   3. (c) John Napier
   4. (d) Euler
   Show answer

   (a) Al-Khwarizmi; Al-Karaji expanded his work later (p. 2).

5. 5. The set of real numbers is:

   1. (a) R = Q ∩ Q′
   2. (b) R = N ∪ Z
   3. (c) R = Q′ only
   4. (d) R = Q ∪ Q′
   Show answer

   (d) The union of rationals and irrationals (p. 3).

Quick revision

• Sumerians used base 60, Egyptians base 10, Romans seven letters.
• Indians gave zero; Arabs took the numerals 0–9 to Europe.
• R = Q ∪ Q′: every real number is rational or irrational.