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Approximate and compare irrational numbers

Lesson

As we have seen before, surds can be manipulated into simpler forms. For example, $\sqrt{250}$250 can be rewritten as $5\sqrt{10}$510.  When we rewrite a surd in this way we do not lose any accuracy, we are simply changing the way we have expressed the surd by finding a factor of $250$250 that is a square number, and then evaluating it. Therefore, we can say that $5\sqrt{10}$510 still has the exact value of $\sqrt{250}$250

However, sometimes we want to give an answer in decimal form to understand how big it is more easily. When we do this, we will need to use a calculator. Try typing $\sqrt{250}$250 or $5\sqrt{10}$510 into a calculator, and you will get $15.8113883$15.8113883. You should note that this is not the exact value! The calculator has given you a rounded number, as it can only display a certain number of digits. In reality, $5\sqrt{10}$510 is an irrational number! We may not need to be this accurate as the calculator though, and we are often asked to round to 2 decimal places. So $\sqrt{250}$250 is almost equivalent to $15.81$15.81. We could write this as $\sqrt{250}$250 ≈ $15.81$15.81

There're also some ways to approximate surds without using a calculator directly. One way is to consider the nearest integer value as a way to check our workings. This is thanks to a rule that states:

if $aa<b,

then $\sqrt{a}<\sqrt{b}$a<b 

This is pretty obvious, so let's see how it can help us to approximate. Let's say we have a surd $\sqrt{40}$40. If we ask ourselves what are the closest square numbers that are bigger and smaller than $40$40, then we'll find that they're $36$36 and $49$49. So then we have $36<40<49$36<40<49, which leads us to say that $\sqrt{36}$36$<$<$\sqrt{40}$40$<$<$\sqrt{49}$49. And if we evaluate that further we get $6$6$<$<$\sqrt{40}$40$<$<$7$7, so we've managed to narrow this surd down to somewhere between $6$6 and $7$7!

What if we wanted to approximate it further and see if we can get a decimal? There's a method for that as well! Once you know what integers the surd lies between, you can find the decimal part by using the following formula:

decimal part of approximation = $\frac{\text{number inside surd - closest smaller square}}{\text{closest bigger square - closest smaller square}}$number inside surd - closest smaller squareclosest bigger square - closest smaller square

This means that, still using our $\sqrt{40}$40 example, the decimal part would be equal to $\frac{40-36}{49-36}$40364936$0.3$0.3 so $\sqrt{40}$40 can be approximated to $6.3$6.3. If you plug this surd into a calculator, you'll see that it is indeed rounded to $6.3$6.3! However this method only works well on larger numbers, and bigger they are the better they'll work! Try and see the difference between using this on say, $\sqrt{2}$2 and $\sqrt{300}$300

Examples

Question 1

Between which two consecutive integers does $\sqrt{74}$74 lie?

  1. Complete the inequality.

    $\editable{}<\sqrt{74}<\editable{}$<74<

question 2

Find the largest value out of the following

A) $2\pi$2π   B) $\sqrt{50}$50    C) $4.21$4.21    D) $\sqrt{49}$49

Think: Let's convert all the numbers to decimals or approximate them using decimals, so that we can compare them. Remember that $\pi$π is approximately $3.14$3.14.

Do:

$2\pi$2π $2\times3.14$2×3.14
  $=$= $6.28$6.28

$\sqrt{50}$50 is bigger than $\sqrt{49}$49 but smaller than $\sqrt{64}$64 so we can say it's between $7$7 and $8$8.

$\sqrt{49}$49 is not actually a surd in that we can evaluate it to $7$7 exactly.

Therefore the biggest value is $\sqrt{50}$50.

 

Question 3

Approximate $\sqrt{95}$95 to the nearest hundredth without using a calculator.

Think: Hundredths are represented by the second decimal place, so we need two decimal places. Since $95$95 is a large number, we can use the formula above to approximate its value.

Do:

$81<95$81<95$<$<$100$100

$\sqrt{81}<\sqrt{95}$81<95$<$<$\sqrt{100}$100

$9<\sqrt{95}$9<95$<$<$10$10

So now we know the surd is equal to $9$9 point something.

For the decimal part:

$\frac{\text{number inside surd - closest smaller square}}{\text{closest bigger square - closest smaller square}}$number inside surd - closest smaller squareclosest bigger square - closest smaller square $=$= $\frac{95-81}{100-81}$958110081
  $=$= $\frac{14}{19}$1419
  $0.74$0.74

Therefore $\sqrt{95}$95 is approximately $9.74$9.74.

 

question 4

Andrea needs fences for all $4$4 sides for each of her $3$3 paddocks. If each paddock is square and has an area of $27$27 m2, how many metres of fencing would she need (to the nearest whole number)?

Think: We need to figure out what the perimeter is for one paddock and then multiply it by the number of paddocks.

Do:

One square paddock is $27$27 m2 so $x^2=27$x2=27 where $x$x is the length of one of its sides. That means $x=\sqrt{27}$x=27.

Using our calculator, this surd is approximately $5.2$5.2.

That means one paddock needs $4\times5.2=20.8$4×5.2=20.8 m of fencing.

Therefore she needs $3\times20.8=62$3×20.8=62 m (nearest whole number) of fencing in total.

Outcomes

NA5-2

Use prime numbers, common factors and multiples, and powers (including square roots)

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