Recurring Decimals, Irrationals and Surds

​Hint: Convert all to fractions

Multiple Choice

Write $$0.\overline{45}$$  as an exact fraction

• $$\frac{45}{100}$$  

• $$\frac{45}{99}$$  

Multiple Choice

What is 0.555... as a fraction

• $$\frac{2.5}{4.5}$$  

• $$\frac{5}{9}$$  

• $$\frac{5}{8}$$  

• $$\frac{2}{3}$$  

Multiple Choice

What is 0.333...

• 3/9

• 1/3

• 33/99

• 9/3

Multiple Choice

Convert to a fraction:

• 346/999

• 115/333

• 75/99

• 34/999

Multiple Choice

What is 10 times 0.666...

• 66.66

• 6/9

• 2/3

• 6.66

Multiple Select

Which two equations do I need to write  $$0.\overline{24}$$  as a fraction

• $$x=0.\overline{24}$$  and  $$100x=24.\overline{24}$$  
  

• $$x=2.\overline{42}$$  and  $$100x=24.\overline{24}$$  
  

Multiple Select

What is different about the decimal  $$0.0\overline{34}$$  

• There is a digit before the recurring digits

• Nothing, the decimal recurs every 034

Multiple Choice

a number that can be expressed as a ratio of two integers

• rational number

• irrational number

• terminating decimal

• repeating decimal

Facts about Rational Numbers

• can be a whole number

• can be a fraction

• can be an integer

• can be a ratio

• can be a repeating decimal

• can be a terminating decimal

• can be a perfect square

Multiple Select

Which numbers are rational numbers?

• 7

• $$\sqrt{64}$$

• 0.5

• $$\sqrt{5}$$

• $$\sqrt{50}$$

Multiple Choice

a number that cannot be written in the form of

$$\frac{a}{b}$$  where a and b are integers and  $$b\ne0$$  

• decimal 

• perfect square

• rational number

• irrational number

Facts about Irrational Numbers

• cannot be written as a fraction or ratio

• non terminating decimal

• non repeating decimal

• non perfect squares

Multiple Select

Which numbers are irrational?

• $$\sqrt{100}$$

• $$\pi$$

• $$\sqrt{4}$$

• $$\sqrt{10}$$

• $$\sqrt{9}$$

A surd is an expression that includes a square root, cube root or other root symbol.

Surds are used to write irrational numbers precisely.

Leaving the square root sign in is useful for two reasons:

- We don´t need calculators with dealing with them

- They are much more accurate as you don´t have to round off the decimal

Multiple Select

For example: A square has an area of 3 $$m^2$$  . 

Indicate the exact length of the side of the square.

• 1,73...... m

• $$\sqrt{3}\ m$$  

This answer is in surd form. It is irrational and it is said to be "in exact form". A decimal answer, such as 1.73 (2 decimal places), is not exact.

Even 1.732050807568877 is not exact. When an answer is required in exact form, you must write it as a surd, ideally simplifying it if possible.

Multiple Select

Which of the following are surds?

• $$\frac{\pi}{2}$$

• $$\sqrt{5}$$

• $$\left(\sqrt{17}\right)^2$$

• $$\sqrt{3}$$

• $$\sqrt{2}+1$$

The following rules apply:

​These are corollaries of the laws of indices

Examples:

Simplifying surds:

Another example:

Multiple Choice

Do you think

$$\sqrt{\left(3\right)\left(4\right)}$$  is equal to  $$\sqrt{3}\times\sqrt{4}$$  ?

• Yes 

• No

Multiple Choice

Simplify: 
√5 x √6

• 2√15
• √11
• √30
• 3√10

Multiple Choice

Simplify fully: 
√32

• 4√8
• 16√2
• 4√2
• 4√4

Multiple Choice

Simplify the following surd...

• 5√5

• 3√5

• 15√3

• 5√3

Multiple Choice

What is a surd?

• A recurring decimal

• An irrational number

• A fraction

• A rational number

Multiple Choice

What is a rational number ?

• A rational number  is a number that cannot be written as a fraction.
• A rational number is a number that can be written as a fraction.

• A rational number cannot be a repeating  decimal.

Multiple Choice

Which of the following is an example of a surd?

• √144

• -4

• 3.3333333333333333...

• √7

Multiple Choice

Simplify the following: a⁰ + a⁰

• a⁰

• 2a⁰

• 1

• 2

Multiple Choice

Evaluate:

• $$\sqrt{4}$$

• $$2$$

• 
  

  $$\frac{1}{2}$$

• $$\sqrt{\frac{1}{4}}$$