Recognise and locate irrational numbers

Number

\(\displaystyle 0.\overline{6}\)

Which statement about the exact number is correct?

A rational number has an exact integer-ratio representation

A rational number can be written as a divided by b, where a and b are integers and b is not zero

The denominator need not divide the numerator evenly. Every quotient of an integer by a positive whole number is rational

Integers, including zero and negative integers, can be written over one

Terminating decimals have power-of-ten denominators; recurring decimals convert to fractions by aligning identical tails

A real number that cannot be represented by any such integer ratio is irrational. Its decimal expansion neither terminates nor eventually repeats

A negative integer quotient is still rational

Is this number rational or irrational?

The numerator is the integer negative seven and the denominator is the non-zero integer five

The exact displayed value is already an integer quotient

It is rational; the negative sign does not change this criterion

A denominator need not divide evenly

Nonterminating does not mean irrational

A finite decimal close to pi is not pi

Classify zero

Use the criterion for positive integer radicands

The principal square root is the non-negative number whose square equals the radicand

For a positive integer N, its square root is rational exactly when N is a perfect square of an integer

This established criterion lets us classify the root; merely seeing many decimal digits would not establish irrationality

Eighteen is not an integer square, so its principal root is irrational and lies between four and five

Do not replace the criterion by the claim that every root sign means irrational. A perfect-square root such as seven is rational

Classify a non-square integer root

Is this number rational or irrational?

The radicand lies between neighbouring integer squares, so it is not a perfect square. By the known non-square integer root criterion, its square root is irrational

Check the radicand against integer squares

A root symbol can represent an integer

Do not extend the integer criterion to arbitrary decimal radicands

The criterion just used concerns positive integer radicands. A non-integer radicand can still have a rational square root

Two point two five is the square of a rational number even though it is not an integer square

Check the exact squaring relationship rather than assuming every decimal radicand gives an irrational root

Root

\(\displaystyle \sqrt{0.81}\)

Find the principal root as an exact decimal

Find a non-negative decimal whose square is exactly 0.81

Recall consecutive whole-number bounds

For non-negative numbers, squaring preserves their order. The nearest integer squares give adjacent whole-number root bounds

Bounds locate a value without replacing it by an exact integer or decimal

Bounds

\(\displaystyle \boxed{\phantom{6}}<\sqrt{41}<7\)

Complete the pair of consecutive whole-number bounds

Compare forty-one with the neighbouring integer squares

Narrow the interval with exact decimal squares

To find adjacent tenths around a positive root, square the nearby tenths exactly

Repeat with hundredths to make a narrower interval

These are enclosing bounds, not a request to round the root. Check both squared inequalities and use adjacent decimals at the requested place value

Construct both enclosing tenths

Give the two adjacent tenths strictly below and above the square root, with the lower bound first

Enter both bounds as decimals, not rounded approximations to the root

The principal square root is positive, so compare squares of positive decimal candidates

The root lies between 1 and 2. Refine this interval with tenths until adjacent candidates have squares on opposite sides of the radicand

Scale each candidate by 10 to check integer squares exactly

Divide by 100 to compare the decimal squares with the radicand

The bounds differ by 0.1, so they are adjacent tenths

These exact inequalities locate the root; neither decimal bound equals the root, and bounds alone do not prove irrationality

Construct adjacent tenths rather than one rounded estimate

Verify both endpoints by exact squaring

Give the two adjacent hundredths strictly below and above the square root, with the lower bound first

Enter both bounds as decimals, not rounded approximations to the root

The principal square root is positive, so compare squares of positive decimal candidates

The root lies between 2 and 3. Refine this interval with hundredths until adjacent candidates have squares on opposite sides of the radicand

Scale each candidate by 100 to check integer squares exactly

Divide by 10000 to compare the decimal squares with the radicand

The bounds differ by 0.01, so they are adjacent hundredths

These exact inequalities locate the root; neither decimal bound equals the root, and bounds alone do not prove irrationality

Narrow the interval to one hundredth

Proposed lower bound for the square root of five

\(\displaystyle 2.23\)

Calculate the square of the proposed lower bound as an exact decimal

Multiply 223 by 223, then account for four decimal places

The exact root has a position even when the drawing is approximate

The marked point lies between the two verified decimal bounds. The drawing uses a finite decimal approximation for its position

The exact value remains the square root of two. It is not exactly any finite coordinate used by the screen

More accurate plotting can narrow the visual uncertainty without making the irrational value rational

A square side need not be a rational length

For a square with positive side length s and area A, including their units, the quantities satisfy

The side length is positive, so take the principal root rather than a negative solution

For an exact area of two square metres, the exact side length is

The value is irrational by the positive-integer non-square criterion. An area of nine square metres instead gives a rational side length of three metres

Area has square units; the side length has length units

Exact side length

\(\displaystyle s=\sqrt{\boxed{\phantom{11}}}\text{ m}\)

A square has an exact area of 11 square metres. Complete its exact side-length expression by entering the radicand only

Undo s squared by taking the principal root of the area measure

Classify the exact side length, not a rounded measurement

Compare the area settings two and nine, then try the other whole-number areas. The picture shows the side length, and the readout gives it to six decimal places; use the integer radicand to decide rationality

What is correct for the area settings two and nine square metres?

Give a usable enclosing interval for a square side

The exact circle ratio is irrational

From earlier circle work, circumference C divided by positive diameter d is the same exact constant for every ideal circle

Pi is known to be irrational. Its irrationality is not proved by a finite measurement, a decimal display or the short investigation here

The rule about integer quotients does not force pi to be rational: the exact circumference and diameter need not both have integer numerical measures

A useful finite approximation is a rational number close to pi, not the exact constant

Name the exact type of the circle constant

Is this number rational or irrational?

Pi is known to have no exact representation as a quotient of two integers with non-zero denominator

It is therefore irrational; this topic uses that established fact rather than proving it

A finite decimal display or a measured circle ratio can approximate pi but cannot prove its irrationality

Which statement is correct?

Locate pi within a stated interval

The following established bounds locate pi between adjacent thousandths

The line marks an approximate location within that interval. The bounds do not claim that pi equals either endpoint

The two endpoint decimals are rational even though the number between them is irrational

A measured ratio is not the exact ideal-circle constant

Suppose a measured circumference is recorded as 31.4 centimetres and its diameter as 10 centimetres

This calculation uses finite recorded measurements, so the resulting ratio is rational

The exact ideal-circle ratio is pi. Ruler resolution, string placement and rounding explain why a measured ratio is only an approximation

A close result supports the model but cannot establish exact equality or prove irrationality

Recorded ratio

\(\displaystyle 62.8\div20\)

A measured circle has recorded circumference 62.8 cm and diameter 20 cm. Calculate the quotient of those recorded values as a decimal

Divide by two, then by ten

Several measured circumference-to-diameter ratios are close to 3.14. What conclusion is justified?

A screen shows 1.7320508 for a calculation but the exact input and rule are hidden. What can you conclude from the display alone?

Record exact forms, estimates and evidence in an independent investigation

This investigation is not automatically marked. Choose three positive whole-number square areas, including a perfect square and a non-square

For each area, write the exact side-length expression with units, classify the value and justify the classification using the radicand

Use a calculator or spreadsheet to estimate each side, then verify an enclosing interval by squaring its decimal endpoints exactly

Plot an approximate position on a labelled number line. Distinguish the exact root from the finite coordinate used in your drawing

Measure circumference and diameter of several circular objects, record units and measurement precision, and compare the calculated ratios

Explain why the measurements support a common ratio but do not prove the irrationality of pi. Do not invent measured data if you cannot perform the measurements

Keep exact classification separate from approximate location

Integer quotients with a non-zero denominator are rational, including terminating and recurring decimal values

For positive integer radicands, perfect-square roots are rational and non-square roots are irrational

Do not extend that criterion carelessly to decimal radicands: a decimal can itself be the square of a rational number

Exact squared inequalities verify enclosing decimal bounds. A plotted position or calculator display is only approximate when the value is irrational

Square-side lengths and the circle ratio pi provide applied irrational values without requiring surd manipulation or geometric construction

Known irrationality criteria and exact definitions justify classification; finite displays and measurements do not prove irrationality