Mathematics · Classes 9-10 · Chapter 1

Classification of Real Numbers.

Is 9 prime? Plenty of people say yes because it is odd. Is 1 prime? Is zero positive? In this note the whole family of real numbers is laid out as one tree, and every number goes into its most specific box.

1. Natural numbers, primes and composites

1, 2, 3, 4, ... are the natural numbers, the positive whole numbers. Among them 2, 3, 5, 7, ... are primes: divisible only by 1 and by themselves, like 11, 13 and 29. And 4, 6, 8, 9, ... are composite: they have other factors too, like 12 = 2 × 6, 30 = 5 × 6 and 39 = 3 × 13.

1 is neither prime nor composite. So in the classification chart the positive integers split three ways: prime, 1 and composite. From 1 to 10 the primes are 2, 3, 5, 7, four in all. The 1 does not count.

If the GCD of two natural numbers is 1, they are called mutually prime (co-prime). Take 6 and 35: the factors of 6 are 1, 2, 3, 6 and of 35 are 1, 5, 7, 35, so the only common factor is 1. Notice that neither 6 nor 35 is prime, yet they are mutually prime.

2. Integers and fractions

All positive and negative whole numbers together with zero are the integers: ..., −3, −2, −1, 0, 1, 2, 3, ... Zero sits in the middle of the list, in a place of its own.

A number of the form p/q is a (simple) fraction where q ≠ 0, q ≠ 1 and p is not exactly divisible by q. For example 1/2, 3/2, −5/3.

If p < q the fraction is proper (1/2, 2/3), and if p > q it is improper (3/2, 5/4). An improper fraction can also be written as a mixed fraction: 14/5 = 2 4/5, because 14 = 2 × 5 + 4. In the chart, improper and mixed are two forms of the same number.

Now all of them together. Put each number in its most specific box; there are eight boxes.

3. Real numbers and their signs

All rational numbers and all irrational numbers together are called the real numbers. What rational and irrational mean comes in the next note. Here, just hold on to this: no real number lies outside these two groups.

Real numbers bigger than 0 are positive; real numbers smaller than 0 are negative. All the positive numbers together with 0 are non-negative. So positive, negative and zero, all three together, make up the real numbers. Zero is neither positive nor negative, yet it is non-negative.

4. The classification chart

Before reading the chart, look at the boxes. Drop a number and it comes to rest in the smallest box that holds it, lighting up every box on the way.

The first step of the book's chart is the shortest one: real numbers split into two, rational and irrational.

Rational numbers split in two again: integers and fractions. Integers split three ways: positive, 0 and negative.

Fractions split in two: simple (proper, improper, mixed) and decimal (finite, infinite repeated). Irrational numbers are infinite decimals without repetition. So decimals come in three kinds in all: finite and repeated sit under rational, infinite non-repeating sits under irrational. Open the tree, one branch at a time.

Now place some numbers in the chart. 3/4 is a proper fraction. 5 is prime. −7 is a negative integer. √13 is irrational. 0 is zero. 1 sits in its own box under positive integers. 9/7 is improper. 12 is composite. 2 4/5 is mixed. 1.1234 is a finite decimal. 0.3̇23̇ is a repeated decimal.

5. Check yourself

Start with the first question.

Think a little before the second.

For the last one, picture the chart.

The whole note on seven cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Classification of Real Numbers” part.