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Number Systems

Study it inNumbers Level 1: What Is a Number?

Map

Arrows point from what you learn first to what it opens. Point at a concept to light up its chain and where it comes back deeper. Right-click a concept (or press and hold it) for its lessons and stories.

  • Proven to depth123
  • Open
  • Locked
  • Comes back deeper
Depth 2Depth 2Comparing numbersComplex numbersCountingIntegersIrrational numbersLarge numbersNatural numbersNumber basesNumeralsPlace valueRational numbersReal numbersThe number lineWhat a number isZeroFractionsIn FractionsPythagoras' theoremIn GeometrySquare rootsIn Arithmetic

Learning path

In the order the concepts build on each other.

  1. CountingReadyCounting is matching things one to one with the counting words in their fixed order; the last word says how many there are. Up to about four things we can see how many at a glance.
  2. Natural numbersLockedThe counting numbers 1, 2, 3, ... - what you get by starting at one and adding one more, again and again. There is no biggest one.Opens after: Counting
  3. NumeralsLockedThe signs we write numbers with. Tally marks need one mark per thing; Roman numerals add up signs for groups; the ten Hindu-Arabic numerals 0 to 9, invented in India, write any number in a few signs.Opens after: Counting
  4. What a number isLockedThree is not three apples and not the numeral 3. It is what every group of three shares: groups that can be matched one to one have the same number (Frege's answer). Numerals and words are only its names. Whether numbers are real things, ideas or a language we invented is still argued about.Opens after: Counting, Numerals
  5. ZeroLockedThe number of things when there is nothing there. Adding or taking away zero changes nothing, and a number take away itself is zero. Brahmagupta, in India in 628, first wrote rules for zero as a number.Opens after: Natural numbers
  6. Place valueLockedIn our numbers, a digit's value depends on its place: ones, tens, hundreds, thousands, each ten times the one to its right. A zero holds an empty place open.Opens after: Numerals, Zero
  7. The number lineLockedA straight line with the numbers marked in order at equal steps, bigger numbers further to the right. Counting is hopping along it, and there is room between the marks for numbers yet to come.Opens after: Natural numbers, Zero
  8. Comparing numbersLockedOf two numbers, the one further right on the number line is greater. We write 7 > 4 and 4 < 7, the wide side of the sign towards the bigger number. With big numbers, more digits means bigger; with as many digits, compare from the left.Opens after: The number line, Place value
  9. Large numbersLockedThousands, millions and billions: each is a thousand times the one before. Counting the zeros, and getting a feel for how big a million really is.Opens after: Place value
  10. Number basesLockedThe size of the bundle that fills a place. Our everyday numbers are base ten; clocks still count minutes and seconds in the Babylonians' sixties, and computers count in base two with only 0 and 1.Opens after: Place value
  11. IntegersLockedThe whole numbers with their negatives: ..., -2, -1, 0, 1, 2, ... Negative numbers sit left of zero and measure things below a starting point, like cold days or money owed. A number plus its opposite is zero, and with negatives every subtraction has an answer.Opens after: The number line, Comparing numbers
  12. Rational numbersLockedNumbers that can be written as a fraction p/q of two integers with q not zero - the numbers you need so that division always has an answer.Opens after: Fractions, Integers
  13. Irrational numbersLockedA number that no fraction can name exactly, like √2, the diagonal of a square with sides of 1, or π. It has its own point on the number line, but its decimals never end and never settle into a pattern that repeats.Opens after: Rational numbers, Pythagoras' theorem, Square roots
  14. Real numbersLockedAll the rational and irrational numbers together - every point on the continuous number line, with no gaps.Opens after: Rational numbers, Irrational numbers
  15. Complex numbersLockedNumbers of the form a + bi, where i is a number with i² = -1. They give every polynomial equation a solution.Opens after: Real numbers

Connections to other domains