UNIT 1 VOCABULARY: RATIONAL AND IRRATIONAL NUMBERS


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1 UNIT VOCABULARY: RATIONAL AND IRRATIONAL NUMBERS 0. How to read fractions? REMEMBER! TERMS OF A FRACTION Fractions are written in the form number b is not 0. The number a is called the numerator, and tells us how many equal parts we have. The number b is called the denominator, and tells us how many equal parts are available. a b where a and b are whole numbers, and the REMEMBER! READING FRACTIONS We use the cardinals to name the numerator and the ordinals for the denominator. For example: > two thirds > seven fifths > one eight 8 Three exceptions: When the denominator is, it is read "half". For example: > one half > three halves When the denominator is, it can be read as "fourth" or "quarter". For example: > a fourth or a quarter > three quarters or three fourths For denominators greater than 0, we say over and do not use ordinal. For example: 5 > twelve over fifteen 7 > seventeen over thirtytwo BE CAREFUL WITH PLURALS! If the numerator is greater than, you must use plurals with ordinals. 5 > five halves > three quarters > seven tenths Exercise. Write in words and read the following fractions: Exercise. Write and read the fractions that represent the shaded portions. Exercise. Which fraction of this window is broken? Which fraction is OK?
2 .. Highest Common Factor To work out the Highest Common Factor (HCF) of several numbers, first you have to find the prime factor decomposition of the given numbers and then, to take and multiply the common factors with the least index. to find out the HCF of numbers 6, 8 y 90: Write them as a product of prime factors: 6= 8= 90= 5 Take and multiply the common factors with the least index: H.C.F. (6,8,90) = = 6.. Lowest Common Multiple To work out the Lowest Common Multiple (LCM) of several numbers, first write them as a product of their prime factors and then take and multiply the common and noncommon factors with the highest index. to find out the HCF of numbers 6, 8 y 90: Write them as a product of prime factors: 6= 8= 90= 5 Take and multiply the common factors with the least index: H.C.F. (6,8,90) = = 6 Exercise. Write a number with 8 digits over the lines and translate the following words:.. Representation of Rational Numbers on the Number Line You have learnt to represent natural numbers, whole numbers an integers on a number line. Natural Numbers: the line extends indefinitely only to the right side of. Whole numbers: the line extends indefinitely to the right, but from 0. Integers: the line extends indefinitely on both sides. Do you see any numbers between, 0; 0, etc.? Rational Numbers: the line extends indefinitely on both sides. But you can now see numbers between, 0; 0, etc. Any rational number can be represented on the number line. In a rational number, the denominator tells us the number of equal parts into which the first unit has been divided; the numerator tells us how many of these parts are considered. So, a rational number such as means four of nine equal 9 7 parts on the right of 0, and for  we make 7 markings of distance each on the left of
3 zero and starting from 0... Equivalent Fractions Equivalent fractions are fractions that look different from each other, but are really the same. We can test if two fractions are equivalent by taking the crossproduct. Exercise. Write the missing numerators or denominators: x 8 = 5 = x 5 7 x =0 5 7 = x.5. Comparing and ordering fractions REMEMBER! SYMBOLS To compare two numbers, we can use these symbols: Symbol Is read Example Is read = Is equal to / equals = Isn't equal to / doesn't equal < Is less than < > Is greater than 5 < 5 6 A half equals two fourths Two thirds doesn't equal three halves Two thirds is less than three quarters Five thirds is greater than five sixths To compare fractions: If the denominators are the same, all you have to do is compare the numerators when it's bigger, the fraction's bigger. 7 and <5 7 If the numerators are the same, all you have to do is compare the denominators when it's smallest, the fraction's smallest.
4 5 and 9 9 < 5 If numerators and denominators are different, you can reduce them to a common denominator. For this, complete the following steps: Find the LCM of the denominators as it is the smallest number that both denominators divide into. We divide every new denominator by the previous one and then we multiply the result by the numerator. 5 and 7 8. It is easy to see that 0 is the LCM, so 5 = [] 0 and In 7 8 = [ ] 0. 5 = [] 0 we must multiply by 8, so we divide the new denominator 0 by the previous one that is 5, what gives us 8, 5 = [] 0. Repeating the process with the second fraction: Finally, we can say that 5 < = Simplest form of a fraction. Simplifying (or reducing) fractions means to make the fraction as simple as possible. Why say foureighths ( 8) when you really mean half ( )? There are two ways to find the simplest form of a fraction: Try dividing both the top and bottom of the fraction until you can't go any further (try dividing by,,5,7,... etc). to reduce 08 to its lowest terms: You can divide both the top and bottom of the fraction by the Greatest Common Factor, (you have to work it out first!). to find the simplest form of The GCF of 8 and is. Divide both top and bottom by. 8 :
5 Exercise. Cancel these fractions to their lowest terms, without using a calculator: 6 = = 8 0 = 50 0 =.. Addition and subtraction with fractions REMEMBER! + PLUS  MINUS To add or subtract fractions: If the fractions have the same denominator, the numerator of the sum or the difference is found by simply adding or subtracting the numerators over the denominator. + = + = 5 = = REDUCE ALWAYS WHEN POSSIBLE!!!! If the denominators are different, follow these steps: Reduce them to a common denominator (see comparing fractions). Add or subtract the numerators and do not change the denominator. Reduce (if possible). to find out The LCM of 6 and is, so = 0 +7 =... Multiplication with fractions REMEMBER! To multiply fractions: Multiply the top numbers (numerators). Multiply the bottom numbers (denominators). Simplify the fraction (if possible). TIMES/MULTIPLIED BY.. Division with fractions REMEMBER! : DIVIDED BY To divide fractions: Multiply the numerator of the first fraction by the denominator of the second. Multiply the denominator of the first fraction by the numerator of the second. Simplify the fraction (if possible).
6 .. Order of Operations When you have several operations to do, which one do you calculate first? We work out operations in this order: BRACKETS EXPONENTS (Powers, roots, etc) DIVISION and MULTIPLICATION (working from left to right) ADDITION and SUBTRACTION (working from left to right) That makes BEDMAS! Exercise. Work out and simplify: 5 = = 7 7 : = = = 5 ( ) = ( 7 7) +: 5 7 = ( 5 ) = = =.. Rational Numbers A rational number is any number which can be written in te form of a fraction. Rational numbers are represented by the letter Q. a So, a rational number can be written as, where a and b are integers and b 0. b But, what numbers can be expressed as a fraction? Rational numbers include: RATIONAL NUMBERS Q Natural Numbers N Integers Z Exact (or terminating) decimal numbers Recurring decimal numbers, with a periodic part. 6= 6 = 0.75= = 5
7 EXERCISE. Match each number with an appropriate sentence... Converting fractions into decimals Every fraction can be expressed as a decimal number. To convert a fraction into a decimal, you just have to divide numerator by denominator. YOU CAN WRITE ANY RATIONAL NUMBER AS A DECIMAL NUMBER, BUT NOT ALL DECIMAL NUMBERS ARE RATIONAL NUMBERS!!! The quotient of a fraction can be: 6 Integers: no decimal part. = Exact (or terminating) decimals: decimal numbers that end (or terminate). 0 =0. Recurring decimal numbers: decimal numbers that have a recurring pattern of a single or multiple digits. =0....=0. = =.58.. Converting decimals into fractions Exact and recurring decimals can be expressed as fractions. To convert a decimal into a fraction, you have to follow these steps: YOU CAN ONLY WRITE RATIONAL NUMBERS AS A FRACTION, THIS IS, JUST EXACT AND RECURRING DECIMALS!!! If a decimal number is exact: The numerator is formed by the digits without the decimal point. The denominator is the number formed by and as many zeros as decimal figures the number has. Reduce, if possible..75= =
8 If a decimal number is recurring: The numerator is the difference between the number formed by the figures including the period (once) and the number formed by the figures BEFORE the period. The denominator is the number formed by as many 9 as decimal figures the period has and as many 0 as decimal figures the nonperiodic part has AFTER the decimal point. Reduce (f possible) = 9 = 8 9 =6. 8= 8 = = 5 Here, the period has one figure (). Here, the period has two figures (8) and the nonperiodic part figure ()... Irrational Numbers Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. FAMOUS IRRATIONAL NUMBERS Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this: (and more...) The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: (and more...) The Golden Ratio is an irrational number. The first few digits look like this: (and more...) Many square roots, cube roots, etc are also irrational numbers. Examples: = (etc) = (etc) BE CAREFUL! = (rational), and 9 = (rational)... so not all roots are irrational. (Check reading exercise)
9 .. Real Numbers Real Numbers include: Rational numbers, and Irrational numbers. In fact a Real Number can be thought of as any point anywhere on the number line:.. Approximations An approximation of a number is a representation of that number that is not exact, but still close enough to be useful. Rounding off a decimal number to a given number of decimal places is the quickest way to approximate a number. if you wanted to round off,6557 to three decimal places, you would: Step : Mark off the required number of decimal places.,65 57 Step : Check the next digit to see if you must round up or round down. Remember: if the next digit is 5 or more, you must round up, and if it is or less, you must round down.,65 57 must be rounded up. Step : Write the final answer.,65 rounded to decimal places. Exercise. Round off the following to decimal places:, , , , , ,
10 .. Absolute and percentage error If a is an approximation of a real value v, the absolute error of the approximation is e a = v a The error is expressed using the real value unit of measure. the real height of Peter is.59 m, but he always says his height is.60 m. This value,.60 m, is an approximation for the real value. As.60 m is not the real length, there is an error of approximation. The absolute error of the approximation can be calculted this way: e a = = = m. The percentage error shows the error as a percent of the exact value. To calculate it you just divide the absolute error by the exact value and make it a percentage: e r = e a v 00 for Peter, we have this percentage error: absolute error e r = 00= real value = 0.5 % Exercise. Find the absolute error and the percentage error in these approximations: Real value Approximation Absolute error Percentage error People at a library 57 people 60 people Weight of the stone 5.86 g 5. g Number of cars cars 00 cars Annual wage 0, ,75.00
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