Number System

The One Thing

The number system classifies numbers into nested sets and uses decimal expansions to distinguish rational numbers from irrational numbers. Real numbers can be represented on the number line, while exponent laws and the square-root spiral provide algebraic and geometric methods for working with powers and square roots.

Definitions and Results

• Natural numbers: The counting numbers $1,2,3,\dots$.
• Whole numbers: The natural numbers together with zero: $0,1,2,3,\dots$.
• Integers: Positive and negative whole numbers, including zero.
• Rational number: A number that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.
• Irrational number: A number that cannot be written as $\frac{p}{q}$. Its decimal expansion is non-terminating and non-recurring.
• Real numbers: All rational and irrational numbers together. Every real number is either rational or irrational; no real number is both.
• Relationship between number sets:

$$N\subset W\subset Z\subset Q\subset R,$$ where $N$, $W$, $Z$, $Q$, and $R$ denote the natural numbers, whole numbers, integers, rational numbers, and real numbers respectively.

• Integer as a rational number: Every integer is rational because any integer $n$ can be written as $\frac{n}{1}$.
• Decimal expansion: The representation of a number using a decimal point and place values.
• Terminating decimal: A decimal expansion that ends after finitely many digits, such as $0.75$.
• Non-terminating recurring decimal: A decimal expansion that continues indefinitely with a repeating pattern, such as $0.333\dots$ or $1.272727\dots$.
• Non-terminating non-recurring decimal: A decimal expansion that continues indefinitely without a repeating pattern, such as $\pi$ or $\sqrt{2}$.
• Decimal criterion for rational numbers: Every terminating decimal and every recurring decimal represents a rational number.
• Denominator criterion: If $\frac{p}{q}$ is in lowest form, its decimal expansion terminates if and only if the prime factors of $q$ are only $2$ and/or $5$. If $q$ contains any other prime factor, the decimal expansion is non-terminating recurring.
• Examples of irrational numbers: $\sqrt{2},\sqrt{3},\sqrt{5}$, and $\pi$ are irrational. The square root of a natural number is irrational when that number is not a perfect square.
• Number line: A line on which real numbers are represented by points in increasing order. Every point represents exactly one real number.
• Density property: Between any two distinct real numbers there are infinitely many real numbers, including infinitely many rational numbers and infinitely many irrational numbers. Consequently, no two distinct real numbers are consecutive.
• Laws of exponents: For a non-zero number $a$,

$$a^m\times a^n=a^{m+n},a^m÷a^n=a^{m-n},(a^m{)}^n=a^{mn}.$$

• Power of a product and quotient: For non-zero $a$ and $b$,

$$(ab{)}^m=a^m{b}^m,{\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m},$$ wherever the expressions are defined.

• Zero and negative exponents: For a non-zero number $a$,

$$a^0=1,a^{-m}=\frac{1}{a^m},$$ where $m$ is a positive integer.

• Rational exponents: For positive $a$,

$$a^{1/n}=\sqrt[n]{a},a^{m/n}={\left(\sqrt[n]{a}\right)}^m.$$

• Square root: For a non-negative number $a$, the principal square root of $a$ is the non-negative number whose square is $a$.
• Square-root spiral: A geometric construction using right triangles with one unit side to represent the lengths $\sqrt{2},\sqrt{3},\sqrt{4}$, and successive square roots.
• Pythagorean relationship in the square-root spiral: If a right triangle has perpendicular sides of lengths $1$ and $\sqrt{n}$, then

$${\text{hypotenuse}}^2=1^2+(\sqrt{n}{)}^2=1+n=n+1.$$ Therefore, its hypotenuse has length $\sqrt{n+1}$.

Worked Methods

1. Classifying a number

• Determine whether the number is a counting number, a whole number, an integer, rational, irrational, or real number.
• Use the inclusion relationship

$$N\subset W\subset Z\subset Q\subset R.$$

• If the number can be expressed as $\frac{p}{q}$, with integers $p,q$ and $q\ne 0$, classify it as rational.
• If it cannot be expressed in this form, classify it as irrational.
• Remember that all rational and irrational numbers are real numbers.

For example, every integer $n$ is rational because $n=\frac{n}{1}$.

2. Determining the type of decimal expansion

• Express the rational number in lowest form:

$$\frac{p}{q}.$$

• Factor the denominator $q$ into primes.
• If the prime factors of $q$ are only $2$ and/or $5$, the decimal expansion terminates.
• If $q$ contains any prime factor other than $2$ or $5$, the decimal expansion is non-terminating recurring.
• Recognise that every terminating or recurring decimal is rational.

The decimal $0.75$ is terminating. The decimals $0.333\dots$ and $1.272727\dots$ are non-terminating recurring. By contrast, $\pi$ and $\sqrt{2}$ have non-terminating, non-recurring decimal expansions and are irrational.

3. Locating a rational number on the number line

• Identify the integers between which the rational number lies.
• Use its denominator to divide the interval into equal parts.
• Count the required number of parts from the appropriate integer.
• Mark the resulting point on the number line.

This method represents rational numbers by partitioning the interval between suitable integers according to the denominator.

4. Comparing irrational numbers

• Obtain suitable decimal approximations, where appropriate.
• Alternatively, compare their squares when the numbers are non-negative.
• Use their positions on the number line to confirm the comparison.
• Do not treat an irrational decimal approximation as an exact terminating decimal.

This method applies to comparisons involving numbers such as $\sqrt{2},\sqrt{3},\sqrt{5}$, and $\pi$.

5. Simplifying expressions using exponent laws

• For multiplication with the same non-zero base, add the exponents:

$$a^m\times a^n=a^{m+n}.$$

• For division with the same non-zero base, subtract the exponents:

$$a^m÷a^n=a^{m-n}.$$

• For a power raised to another power, multiply the exponents:

$$(a^m{)}^n=a^{mn}.$$

• For a product or quotient raised to a power, apply the power to each factor:

$$(ab{)}^m=a^m{b}^m,{\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}.$$

• Replace a zero exponent by $1$, provided the base is non-zero:

$$a^0=1.$$

• Rewrite a negative exponent using a reciprocal:

$$a^{-m}=\frac{1}{a^m}.$$

• Rewrite rational exponents using roots:

$$a^{1/n}=\sqrt[n]{a},a^{m/n}={\left(\sqrt[n]{a}\right)}^m.$$

6. Constructing square roots using the square-root spiral

• Construct a right triangle with perpendicular sides of lengths $1$ and $1$.
• By Pythagoras’ theorem, the hypotenuse has length

$$\sqrt{1^2+1^2}=\sqrt{2}.$$

• Use the existing hypotenuse of length $\sqrt{2}$ as one perpendicular side of a new right triangle, with the other perpendicular side of length $1$.
• The new hypotenuse has length

$$\sqrt{1^2+(\sqrt{2}{)}^2}=\sqrt{3}.$$

• Repeat the construction. A triangle with perpendicular sides $1$ and $\sqrt{n}$ has hypotenuse

$$\sqrt{1^2+(\sqrt{n}{)}^2}=\sqrt{n+1}.$$

• Transfer each hypotenuse length to the number line to represent $\sqrt{2},\sqrt{3},\sqrt{4}$, and successive square roots.

Where It Goes Wrong

• Forgetting that the rational-number test applies to $\frac{p}{q}$ in lowest form; the denominator must be reduced before its prime factors are examined.
• Treating a non-terminating decimal as automatically irrational; recurring decimals such as $0.333\dots$ and $1.272727\dots$ are rational.
• Confusing non-terminating recurring decimals with non-terminating non-recurring decimals; the latter, such as $\pi$ and $\sqrt{2}$, are irrational.
• Omitting the condition $q\ne 0$ when defining a rational number, or applying exponent laws without the required non-zero-base conditions.
• Using $a^0=1$ when $a=0$; the stated rule requires a non-zero base.
• Forgetting that the principal square root is non-negative, or omitting the unit side and the Pythagorean relationship in the square-root spiral.

What Gets Asked

• Classify given numbers as natural, whole, integer, rational, irrational, or real numbers.
• State or use the inclusion relationship

$$N\subset W\subset Z\subset Q\subset R.$$

• Determine whether a decimal expansion is terminating, non-terminating recurring, or non-terminating non-recurring.
• Decide whether a rational number has a terminating decimal expansion by examining the prime factors of its denominator in lowest form.
• Explain why every integer is rational.
• Identify examples of irrational numbers, including $\sqrt{2},\sqrt{3},\sqrt{5}$, and $\pi$.
• Locate rational numbers on the number line.
• Compare irrational numbers using approximations, squares, or number-line positions.
• Use the density property to identify infinitely many rational and irrational numbers between two distinct real numbers.
• Simplify powers using the laws of exponents, including zero, negative, and rational exponents.
• Construct or explain the square-root spiral and use

$${\text{hypotenuse}}^2=\text{sum of the squares of the perpendicular sides}$$ to obtain $\sqrt{2},\sqrt{3},\sqrt{4}$, and successive square roots.