Chapter 1.1 — Introduction to Real Numbers

Placing a Rational Number on the Number Line

Any rational number occupies one exact spot on the number line. The method is to convert it to a mixed number, identify the two consecutive integers it falls between, and divide that unit segment into as many equal parts as the denominator.

Take 7/4. Since 7/4 = 1¾, it sits between 1 and 2. Splitting that interval into 4 equal parts places 7/4 at the third mark past 1. The negative case works identically on the other side: −7/4 = −1¾ sits between −1 and −2, three-quarters of the way from −1 toward −2.

A second example makes the pattern stick: locate −11/3. As a mixed number, −11/3 = −3⅔, so it sits between −3 and −4, at the second of three equal divisions past −3 going in the negative direction. Whatever the denominator turns out to be, that number becomes the number of equal parts the relevant unit segment is split into — there's no need to memorise separate rules for positive and negative fractions, only to track which pair of integers the value falls between.

Squeezing Infinitely Many Rationals Between Two Numbers

Between any two rational numbers, no matter how close together, infinitely many other rational numbers exist. Two methods make this concrete rather than just theoretical.

The mean method: the average (a + b)/2 of two rational numbers always lies between them. Repeating the process — taking the mean of the new value and one of the original endpoints — generates as many more as needed. The mean of 4 and 5 is 9/2; the mean of 4 and 9/2 is 17/4, giving the chain 4 < 17/4 < 9/2 < 5.

The common-denominator method is faster when several numbers are needed at once. To insert n rational numbers between two integers, rewrite both as fractions over (n + 1) and read off everything in between. Inserting 7 numbers between 4 and 5 means using denominator 8: 4 = 32/8 and 5 = 40/8, so 33/8 through 39/8 are the seven required numbers, all confirmed by the chain 32/8 < 33/8 < ⋯ < 39/8 < 40/8.

Nothing about either method depends on the two starting numbers being whole numbers or being close together — the same steps work for 3/5 and 2/3 just as well as for 4 and 5, provided the denominators are equalised first. That's really the point of this section: there is no smallest gap between two distinct rational numbers, no matter how the two are chosen. Halve the gap, halve it again, and a new rational number appears every time — which is what mathematicians mean when they describe the rationals as dense on the number line.

Why Every Rational Number's Decimal Either Stops or Repeats

Dividing p by q for any rational number produces a decimal, and that decimal only ever behaves in one of two ways — there is no third option.

• Terminating: the division reaches a remainder of zero and stops. 3/8 = 0.375.
• Non-terminating, recurring: a block of digits repeats forever without ever hitting a zero remainder. 2/11 = 0.181818… = 0.1̄8̄, and 7/6 = 1.1666… = 1.16̄.

Reading the Denominator to Predict the Decimal, Without Dividing

There's a shortcut that avoids doing the division at all: look only at the prime factorisation of the denominator, once the fraction is in lowest terms.

• If the only prime factors present are 2 and 5 — that is, the denominator has the form 2^m × 5ⁿ — the decimal terminates. 3/8 has denominator 2³, giving 0.375; 9/25 has denominator 5², giving 0.36.
• If any other prime factor appears in the denominator, the decimal is non-terminating and recurring. 5/9 has denominator 3², giving 0.5̄; 8/15 has denominator 3 × 5, giving 0.53̄.

The reason traces back to how decimals work: a terminating decimal is always some whole number over a power of 10, and 10 factors only as 2 × 5. A denominator that already fits the 2^m × 5ⁿ pattern can always be scaled up to become an exact power of 10 by multiplying top and bottom by whatever 2s or 5s are missing — no other prime can do that.

Try it on 7/20. The denominator 20 = 2² × 5¹ is missing one extra factor of 5 to balance the powers of 2 and 5, so multiplying top and bottom by 5 gives 35/100 = 0.35 — terminating, exactly as predicted, and without a single division step. Compare that with 7/12: the denominator 12 = 2² × 3 carries a factor of 3 that no amount of multiplying by 2s or 5s can remove, so no power-of-10 denominator is reachable, and the decimal must recur instead.

Reversing the Process — Decimal Back to p/q

Converting a terminating decimal back to a fraction just means writing the digits over the matching power of 10 and simplifying: 0.375 = 375/1000 = 3/8. A purely recurring decimal goes over the same number of 9s as there are repeating digits: 0.1̄8̄ = 18/99 = 2/11. A mixed decimal like 1.16̄, where only part of it repeats, needs the non-repeating part subtracted out first — working through it gives 1.16̄ = (116 − 11)/90 = 105/90 = 7/6.

The subtraction step is easy to lose track of, so it's worth seeing why it works rather than memorising it. Let x = 1.16̄ = 1.1666…. Multiplying by 10 gives 10x = 11.666…, and multiplying by 100 gives 100x = 116.666…. Subtracting the first from the second cancels the entire infinite repeating tail: 100x − 10x = 116.666… − 11.666… = 105, so 90x = 105 and x = 105/90 = 7/6 — the same answer as the shortcut, but with the reasoning made visible.

Where the Chapter Goes From Here

This introduction sets up everything that follows in Chapter 1. Exercise 1.1 drills these exact skills — classifying numbers, inserting rationals between values, and converting decimals both ways. From there, Exercise 1.2 asks what happens to numbers whose decimals never settle into a repeating pattern at all, which is where irrational numbers enter the picture. The terminating/recurring rule resurfaces later in Class 10's decimal expansion work, this time derived from the Fundamental Theorem of Arithmetic rather than long division alone.