STRAND A: NUMBER. UNIT A6 Number Systems: Text. Contents. Section. A6.1 Roman Numerals. A6.2 Number Classification* A6.

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1 STRAND A: NUMBER A6 Number Systems Text Contents Section A6.1 Roman Numerals A6. Number Classification* A6.3 Binary Numbers* A6.4 Adding and Subtracting Binary Numbers*

2 A6 Number Systems A6.1 Roman Numerals The number system that we use in daily life is the base 10 (Arabic) system. Roman numerals, the numeric system used in ancient Rome, uses letters from the Latin alphabet in an equivalent system of representing numbers. In the Roman system we use I for 1 V for 5 X for 10 L for 50 C for 100 D for 500 M for 1000 The values of consecutive letters are ADDED unless a letter with a lower value appears in front of a letter with a higher value. When this is the case, the lower value letter is SUBTRACTED from the higher value, as shown in the following examples: placing I before V or before X to make the numbers 4 (IV) or 9 (IX) placing X before L or C to make the numbers 40 (XL) or 90 (XC) placing C before D or M to make 400 (CD) or 900 (CM). Examples of the effects of the order in which letters are placed: VII = = 7 XXV = = 5 IV = 5 1= 4 IC = = 99 VL = 50 5 = 45 1

3 A6.1 Example 1 Complete these number lines using Roman numerals. (a) I V X (b) X L C (a) I II III IV V VI VII VIII IX X XI XII (b) X XX XXX XL L LX LXX LXXX XC C CX CXX Example Convert the following Roman numerals to base 10. (a) II (b) IX (c) DI (d) LIX (a) 11 = 1+ 1= (b) IX = 10 1 = 9 (c) DI = = 501 (d) LIX = 50 + ( 10 1) = 59 Example 3 Convert the following base 10 numbers to Roman numerals. (a) 14 (b) 84 (c) 7 (d) 1010

4 A6.1 (a) 14 = = 10 + ( 5 1) = XIV 84 = = ( 5 1) = LXXXIV 7 = = XXVII 1010 = = MX Exercises 1. Convert the following Roman numerals to base 10. (a) VIII (b) CD (c) DVI (d) CDXC. Convert the following base 10 numbers to Roman numerals. (a) 6 (b) 109 (c) 56 (d) Complete the number line using Roman numerals. C D M Change the Roman numerals to base 10 numbers. (a) (i) DIX (ii) MCMXLV (iii) CMIV (iv) CDXVI (v) MCXI (vi) CMXCIX (b) (i) List the numbers in (a) above in decreasing order. (ii) Divide the second number in your list by 11 and write the answer in Roman numerals. 5. Above the entrance to a church there is a Roman number MDCCXCI When do you think the church was built? The crypt was built 153 years before the main church. What Roman number is carved in the wall of the crypt? 3

5 A Which Roman numeral can written instead of the shapes to make the statements true? (a) CDLXXIX < < CDLXXXIII (b) CMXCVIII < < MIV Give your answers in Roman numerals. 7. Continue the sequences, giving the next 3 terms, using Roman numerals. (a) XLVII, LXVII, LXXXVII,...,...,..., (b) CMI, DCCCI, DCCI,...,...,..., 8. The sum of any two adjacent numbers is the number directly above them. Fill in the missing numbers. (a) (b) XC L XL XX LXXV L CCCL C A6. Number Classification* The number system is classified into various categories. Natural numbers (or 'counting numbers') These are the positive whole numbers, namely 1,, 3,,,... Integers These are the set of positive and negative whole numbers, e.g. 1, 10, 364, 7, 10. The set of integers is denoted by Z. Rational Numbers A rational number is a number which can be written in the form m, where m and n are integers. n For example, 4 5 is a rational number. A rational number is in its simplest form if m and n have no common factor and n is positive. The set of rational numbers is denoted by Q. 4

6 A6. Irrational Numbers There are numbers which cannot be written in the form m, where m and n are integers. n Examples of irrational numbers are π,, 3, 5. Terminating Decimals These are decimal numbers which stop after a certain number of decimal places. For example, 7 = is a terminating decimal because it stops (terminates) after 3 decimal places. Recurring Decimals These are decimal numbers which keep repeating a digit or group of digits; for example = is a recurring decimal. The six digits repeat in this order. Recurring decimals are written with dots over the first and last digit of the repeating digits, e.g Note All terminating and recurring decimals can be written in the form m, so they are n rational numbers. Real Numbers These are made up of all possible rational and irrational numbers; the set of real numbers is denoted by R. We can use a Venn diagram to illustrate this classification of number, as shown below. R 1+ Q + Q π π N Z 3 3 5

7 A6. R : Real numbers N : Natural numbers Z : Integers Q : Rational numbers Q + : Positive rational numbers Some numbers have been inserted to illustrate each set. Worked Example 1 Classify the following numbers as integers, rational, irrational, recurring decimals, terminating decimals. 5, 7, 0.6, , 8, 11, 10, π 4, 49 Number Rational Irrational Integer Recurring Terminating Decimal Decimal π 4 49 Worked Example Show that the numbers and are rational. 6

8 A6. For a terminating decimal the proof is straightforward = = which is rational because m = 69 and n = 00 are integers. For a recurring decimal, we multiply by a power of ten so that after the decimal point we have only the repeating digits = Also = Subtracting the second equation from the first gives = 917 which is rational because m = 917 and n = Worked Example ( ) = = Prove that 3 is irrational. Assume that 3 is rational so we can find integers m and n such that 3 = m n, and m and n have no common factors. Square both sides, 3 = m n Multiply both sides by n, m = 3n Since 3n is divisible by 3, then m is divisible by 3. Since m is an integer, m is an integer and 3 is prime, then m must be divisible by 3. Let m = 3 p, where p is an integer. 3n = m = ( 3p) = 9p n = 3p Since 3p is divisible by 3, n is divisible by 3 and hence n is divisible by 3. We have shown that m and n are both divisible by 3. This contradicts the original assumption that 3 = m, where m and n are integers with n no factor common. Our original assumption is wrong. 3 is not rational it is irrational. 7

9 A6. Exercises 1. Classify the following numbers as rational or irrational, terminating or recurring decimals. (a) 100 (b) 06. (c) π 0. (d) (e) 075. (f) 1 π (g) 11 (h) (i) π (j) 5 11 (k) sin 45 (l) cos60. Where possible, write each of the following numbers in the form and n are integers with no common factors (a) 0.49 (b) 03. (c) 49 4 (e) (f) 01. (g) 009. (i) 0.15 (j) 0.96 (d) 7 (h) m n, where m Write each of the following recurring decimals in the form integers with no common factors. m, where m and n are n (a) 041 (b) (c) (d) 08. (e) (f) 05. (g) Given the rational numbers show that a a = 3, b = 7 9, c = 11 15, d = b, c+ d, a+ c+ d, ab, cd, bc and abc are rational numbers. 5. Prove that the following numbers are irrational,, 5, What happens if you try to prove that 4 is irrational using the proof by a contradiction method. 8

10 A6. 7. Write down two rational numbers and two irrational numbers which lie in each of the following intervals. (a) 0< x < 1 (b) 1< x < 9 (c) 4< x < (d) 98 < x < Which of these expressions can be written as recurring decimals and which can be written as non-recurring decimals? 7, π, 1 17, 7, x = a + b State whether x is rational or irrational in each of the following cases, and show sufficient working to justify each answer. (a) a = 5 and b = 1 (b) a = 5 and b = 6 (c) a = and b = 7 (d) a = 3 7 and b = (a) Write down two irrational numbers that multiply together to give a rational number. (b) Which of the following numbers are rational? 1 1,, 4, 4 1, π, π 11. (a) A Mathematics student attempted to define an irrational number as follows: "An irrational number is a number which, in its decimal form, goes on and on." (i) Give an example to show that this definition is not correct. (ii) What must be added to the definition to make it correct? (b) Which of the following are rational and which are irrational? 4 1, 6 1, , 1 3 Express each of the rational numbers in the form integers. 3 p, where p and q are q 1. (a) Write down an irrational number that lies between 4 and 5. (b) N is a rational number which is not equal to zero. Show clearly why 1 N must also be rational. 13. n is a positive integer such that n = correct to 1 decimal place. (a) (i) Find a value of n. (b) (ii) Explain why n is irrational. Write down a number between 10 and 11 that has a rational square root. 9

11 A6.3 Binary Numbers* We normally work with numbers in base 10. In this section we consider numbers in base, often called binary numbers. In base 10 we use the digits 0, 1,, 3, 4, 5, 6, 7, 8, and 9. In base we use only the digits 0 and 1. Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents no current flowing whereas 1 represents a current flowing. In base 10 we use a system of place values as shown below: Note that, to obtain the place value for the next digit to the left, we multiply by 10. If we were to add another digit to the front (left) of the numbers above, that number would represent s. In base we use a system of place values as shown below: = = 73 Note that the place values begin with 1 and are multiplied by as you move to the left. Once you know how the place value system works, you can convert binary numbers to base 10, and vice versa. Worked Example 1 Convert the following binary numbers to base 10: (a) 111 (b) 101 (c) For each number, consider the place value of every digit. (a) = 7 The binary number 111 is 7 in base 10. (b) = 5 The binary number 101 is 5 in base

12 A6.3 (c) = 10 The binary number is 10 in base 10. Worked Example Convert the following base 10 numbers into binary numbers: (a) 3 (b) 11 (c) 140 We need to write these numbers in terms of the binary place value headings 1,, 4, 8, 16, 3, 64, 18,..., etc. (a) 1 3 = The base 10 number 3 is written as 11 in base. (b) = The base 10 number 11 is written as 1011 in base. (c) = The base 10 number 140 is written as in base. Exercises 1. Convert the following binary numbers to base 10: (a) 110 (b) 1111 (c) 1001 (d) 1101 (e) (f) (g) (h) (i) (j) (k) (l) Convert the following base 10 numbers to binary numbers: (a) 9 (b) 8 (c) 14 (d) 17 (e) 18 (f) 30 (g) 47 (h) 5 (i) 67 (j) 84 (k) 00 (l)

13 A Convert the following base 10 numbers to binary numbers: (a) 5 (b) 9 (c) 17 (d) 33 Describe any pattern that you notice in these binary numbers. What will be the next base 10 number that will fit this pattern? 4. Convert the following base 10 numbers to binary numbers: (a) 3 (b) 7 (c) 15 (d) 31 What is the next base 10 number that will continue your binary pattern? 5. A particular binary number has 3 digits. (a) What are the largest and smallest possible binary numbers? (b) Convert these numbers to base When a particular base 10 number is converted it gives a 4-digit binary number. What could the original base 10 number be? 7. A 4-digit binary number has zeros and ones. (a) List all the possible binary numbers with these digits. (b) Convert these numbers to base A binary number has 8 digits and is to be converted to base 10. (a) (b) What is the largest possible base 10 answer? What is the smallest possible base 10 answer? 9. The base 10 number 999 is to be converted to binary. How many more digits does the binary number have than the number in base 10? 10. Calculate the difference between the base 10 number and the binary number 11111, giving your answer in base 10. 1

14 A6.4 Adding and Subtracting Binary Numbers It is possible to add and subtract binary numbers in a similar way to base 10 numbers. For example, = 3 in base 10 becomes = 11 in binary. In the same way, 3 1= in base 10 becomes 11 1 = 10 in binary. When you add and subtract binary numbers you will need to be careful when 'carrying' or 'borrowing' as these will take place more often. * Key Addition Results for Binary Numbers = = = 11 Key Subtraction Results for Binary Numbers 1 0 = = = 10 Worked Example 1 Calculate, using binary numbers: (a) (b) (c) (a) 111 (b) 101 (c) Note how important it is to 'carry' correctly. Worked Example Calculate the binary numbers: (a) (b) (c) (a) 111 (b) 110 (c)

15 A6.4 Exercises 1. Calculate the binary numbers: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Calculate the binary numbers: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Calculate the binary numbers: (a) (b) (c) (d) Describe any patterns that you observe in your answers. 4. Calculate the binary numbers: (a) (b) (c) (d) Describe any patterns that you observe in your answers. 5. Solve the following equations, where all numbers, including x, are binary: (a) x + 11 = 1101 (b) x 10 = 101 (c) x 1101 = (d) x = (e) x = (f) x 1001 = Calculate the binary numbers: (a) 10 1 (b) (c) (d) Describe any patterns that you observe in your answers. 14

16 A (a) Convert the binary numbers and 1110 to base 10. (b) (c) (d) Add together the two base 10 numbers. Add together the two binary numbers. Convert your answer to base 10 and compare with your answer to (b). 8. (a) Convert the binary numbers and to base 10. (b) Calculate the difference between the two base 10 numbers. (c) Convert your answer to (b) into a binary number. (d) Calculate the difference between the two binary numbers and compare with your answer to (c). 9. Here are 3 binary numbers: Working in binary, (a) add together the two smaller numbers, (b) add together the two larger numbers, (c) take the smallest number away from the largest number, (d) add together all three numbers. 10. Calculate the binary numbers: (a) (b)

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