© T Madas. Add zero and negative integers Add fractions and decimals which can be represented by fractions Add imaginary numbers Real Numbers ( ) Integers.

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Presentation transcript:

© T Madas

Add zero and negative integers Add fractions and decimals which can be represented by fractions Add imaginary numbers Real Numbers ( ) Integers ( ) E.g. … -3, -2, -1, 0,1, 2, 3 … Rational Numbers ( ) E.g. 17, -10, -½, -0.65, etc Natural Numbers ( ) E.g. 1, 2, 3 … Complex Numbers ( ) Add irrational numbers such as the square roots and π

© T Madas Add zero and negative integers Add fractions and decimals which can be represented by fractions Add imaginary numbers Real Numbers ( ) Integers ( ) E.g. … -3, -2, -1, 0,1, 2, 3 … Rational Numbers ( ) E.g. 17, -10, -½, -0.65, etc Natural Numbers ( ) E.g. 1, 2, 3 … Complex Numbers ( ) Add irrational numbers such as the square roots and π

© T Madas Rationalor Irrational? It doesn’t quite work like this with numbers

© T Madas Rationalor Irrational? It doesn’t quite work like this with numbers

© T Madas What is an irrational number? It is a number which when written in decimal form: it has an infinite number of decimal digits which appear with no pattern whatsoever The formal definition of an irrational number: It is a number that cannot be written as a fraction with integer numerator and denominator The constant ratio of a circle’s circumference is an irrational number, which is given approximately by: [ 40 d.p. ] We know this irrational number as π Infinite decimal places, no pattern, there is no fraction equal to π

π ≈ 1000 decimal places

© T Madas π ≈ 5000 decimal places

© T Madas π ≈ decimal places Never ends It has no pattern No fraction can be found equal to π π is the most famous irrational number

© T Madas All non exact roots are irrational numbers FACT [it can be proven using more advanced maths ] rational irrational rational Why? rational

© T Madas 2 ≈ 1000 decimal places

© T Madas 2 ≈ 5000 decimal places

© T Madas 2 ≈ decimal places Never ends It has no pattern No fraction can be found equal to 2 All non exact roots are irrational numbers

© T Madas All non exact trigonometric ratios are irrational numbers FACT [it can be proven using more advanced maths ] sin 30° = rational irrational tan 60° = cos 71° = irrational rational tan 45° = sin 10° = irrational cos 120° =

© T Madas sin45° ≈ 1000 decimal places

© T Madas sin45° ≈ 5000 decimal places

© T Madas Never ends It has no pattern No fraction can be found equal to sin 45° All non exact trigonometric ratios are irrational numbers sin45° ≈ 1000 decimal places

© T Madas Look at the 500 decimal places of this number Is it rational or irrational?

© T Madas Look at the 500 decimal places of this number Is it rational or irrational?

© T Madas Look at the 500 decimal places of this number Is it rational or irrational?

© T Madas Look at the 500 decimal places of this number Is it rational or irrational?

© T Madas Look at the 500 decimal places of this number Is it rational or irrational? This decimal has a 60 digit recurring pattern Recurring decimals are not irrational numbers Recurring decimals can always be written as fractions 1 61

© T Madas An irrational number, when written in decimal form, it has an infinite number of decimal digits with no pattern whatsoever. or more formally: It is a number that cannot be written as a fraction with integer numerator and denominator Summary on Irrational Numbers Examples of irrational numbers: π all non exact roots all non exact trigonometric ratios NOTE Recurring decimals are rational numbers

© T Madas An irrational number, is a number that cannot be expressed as a fraction with integer numerator and denominator Loosely this means: when an irrational number is written in decimal form, it has an infinite number of decimal digits with no pattern whatsoever. Summary on Irrational Numbers Examples of irrational numbers are: π non exact roots non exact trigonometric ratios or inverse trigonometric ratios e (and most exponentials involving e ) Non exact logarithms to any base Definite integrals which cannot be integrated in terms of known functions

© T Madas

Suppose we have two numbers and only one of them is irrational. Their sum/difference will be irrational Their product/quotient will be irrational Then: E.g.

© T Madas the irrational number with zero whole part and decimal part such that when it is added to 2, it gives 1.999… When both numbers are irrational, things are not always straightforward. Suppose we have two numbers and only one of them is irrational. Their sum/difference will be irrational Their product/quotient will be irrational Then: is irrational BUT is rational

© T Madas When both numbers are irrational, things are not always straightforward. Suppose we have two numbers and only one of them is irrational. Their sum/difference will be irrational Their product/quotient will be irrational Then: is irrational BUT is rational

© T Madas It gets even more complicated with irrational powers Is π π rational or irrational? It is almost certain to be irrational. Today’s mathematical software running on super-fast computers can calculate billions of decimal places, but no one has actually proved this fact to this date. Here is an interesting example of an irrational number raised to an irrational power being rational! is either rational (R ) or irrational (I ) if it is rational (R ) we found our number !!! if not: !!!!

© T Madas