Transition Year Maths 1. Number Systems


We use a base 10 system with 10 digits, they are : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is the decimal place-value system. 437 means 7 × 100 You are all familiar with this system.


(7) plus 3 × 101 (30) plus 4 × 102 (400).


The second system you are likely to encounter is the Roman Numeral system. It does not have place-value. The letters have fixed values and are ordered from largest to smallest. If a letter representing a smaller value appears before a larger one (e.g. IV), the smaller value is subtracted from the larger value (e.g. 5 – 1 = 4, IV).


I = 1; V = 5; X = 10; L = 50; C = 100; D = 500; M = 1000


Example 1: Write out the first 10 numbers in Roman Numerals. I, II, III, IV, V, VI, VII, VIII, IX, X Note: 4 is IV, one before five


Example 2: Write out the Roman Numerals for the following (a) 21 = XXI (b) 900 = CM (c) 1954 = MCMLIV (d) 3592 = MMMDXCII


It is difficult to do calculations using this system without converting back to our base 10 system first.


The third system you might encounter is the Base 2 or Binary System. It uses just two digits: 0 and 1. This is a place value number system.


The number 1101112 means (looking at the digits from right to left) 1 × 20 plus 1 × 21


(210) plus 1 × 22 (410) plus 0 × 23 (010) plus 1 × 24 (1610) plus 1 × 25 1 + 2 + 4 + 0 + 16 + 32 = 5510 answer: 1101112 = 5510


All digital information is carried using this system. The bases for all the technology in your mobile phones, computers and digital TVs is dependent on the binary system. The two digits can be 1, 0 or on, off, or +, – or 6 volts, 0 volts etc. Electronically it is easy to represent the binary system.


A subscript is used to identify the base number. The subscript (e.g. 10) is only used when discussing bases. 1410 is normally just written 14.


Example 3: Write the first 20 numbers in binary 110 =12


210 =102 310 =112 410 = 1002 510 = 1012 610 = 1102 710 = 1112


810 = 10002 910 = 10012 1010 = 10102 1110 = 10112 1210 = 11002 1310 = 11012 1410 = 11102


12


1510 = 11112 1610 = 100002 1710 = 100012 1810 = 100102 1910 = 100112 2010 = 101002


(110) (3210)


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