Search topics

Search

Locate exact square roots on the real number line

Start a set of 8
1 OF 11

Which interval contains the exact positive square root?

\[\sqrt{13}\]
Year 9, Chapter 6, 4 difficulty levels, Calculator allowedBuild a worksheet

Locating real roots

Bounds identify the interval; a geometric construction transfers an exact distance. Calculator decimals approximate that position

  1. Bound a positive root Find consecutive integer squares either side of the radicand. For example, 21 lies between the consecutive squares 16 and 25, giving the interval below \(\displaystyle 4<\sqrt{21}<5\)
  2. Reflect a negative root Negating reverses the endpoint order on the number line \(\displaystyle -5<-\sqrt{21}<-4\)
  3. Keep exact and approximate separate The square root of a non-square positive integer is irrational. A rounded finite decimal is a rational approximation, not the exact value \(\displaystyle \sqrt{21}\approx4.58\)
  4. Transfer a length without changing it Use perpendicular triangle legs to find a hypotenuse. A compass centred at zero transfers that length to the positive number line \(\displaystyle OB^2=1^2+1^2=2\)

Worked example

A right triangle has perpendicular legs of 3 and 1 units. A compass centred at zero transfers its hypotenuse to the positive number line. Locate the resulting point P.

A right triangle has horizontal leg 3 units and vertical leg 1 units. A compass centred at O transfers hypotenuse OB to P on the positive number line. OP and OB have equal length.31OBPMagnified number-line detail: P is just to the right of 3. Close-up near 3 3 P
Lengths are in units. O marks zero. The compass arc preserves the radius; each tick is one unit
\[OB^2=3^2+1^2\]
  1. \(\displaystyle OB^2=10\)Apply Pythagoras to the perpendicular legs
  2. \(\displaystyle OP=OB=\sqrt{10}\)The fixed compass radius preserves the hypotenuse length
  3. \(\displaystyle 9<10<16\)Compare with neighbouring perfect squares
  4. \(\displaystyle 3<\sqrt{10}<4\)P is between 3 and 4; its exact coordinate is √10

Common mistakes

Equating an approximation with the exact root
✕\(\displaystyle \sqrt{13}=3.61\)✓\(\displaystyle \sqrt{13}\approx3.61\)
The finite decimal is rational; the exact square root of 13 is irrational
Keeping the positive endpoint order after negating
✕\(\displaystyle -4<-\sqrt{21}<-5\)✓\(\displaystyle -5<-\sqrt{21}<-4\)
Reflection in zero reverses left and right
Adding lengths instead of squared lengths
✕\(\displaystyle OB=3+1=4\)✓\(\displaystyle OB=\sqrt{3^2+1^2}\)
The hypotenuse comes from Pythagoras. A compass then transfers that unchanged length

Difficulty levels

introLocate positive and negative roots between consecutive integers
standardFind consecutive tenth bounds and round roots to two decimal places
challengeUse a right triangle and compass transfer to locate an exact root
examDistinguish exact and approximate values and compare a root with a rational number

Questions about this topic

Can an irrational number be located exactly?+

Yes. It is a real number with an exact position. Some irrational lengths can be constructed geometrically, even though their decimals do not terminate or repeat

Why does the compass arc preserve the length?+

Every point on a circle is the same distance from its centre. Keep the compass opening fixed at the hypotenuse length

Does squaring always preserve order?+

It preserves order for nonnegative numbers. Handle signs before comparing squares; for example, negative five is less than negative three, but its square, 25, is greater than 9.

Is the drawing a proof that a decimal is exact?+

No. The integer leg lengths and Pythagorean calculation establish the exact root. Coordinates and pixels on a screen are only a display approximation

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 6: Indices and surds

In this chapter

Work with index notationExpress a number in prime-factor index formSimplify using index laws 1 and 2Simplify using index law 3 and the zero powerExpand using index laws 4 and 5Work with negative indicesSolve an exponential equation by matching basesCombine positive and negative integer indicesScientific notationMultiply and divide in scientific notationSI prefixes and scientific timescalesRound to decimal places and significant figuresRecord measurements and compare instrumentsRelate fractional indices to roots (optional extension)Locate exact square roots on the real number lineSimplify, compare and operate with surds (optional practice)