1. Number Systems NOTE


The numbers just go up in order. The next number that can be written after 1 using just 1 and 0 must be 10, the next 11 and so on.


Example 4:


0 units = 0 1 × 21 1 × 22 0 × 23 1 × 24


Convert 101102 to base 10 = 0


=1 ×2= 2 =1 ×4= 4 =0 ×8= 0


=1 × 16 = 16 2210


answer: 101102 = 2210


To convert from base 10 to binary, divide the decimal by 2 and note the remainder. Keep dividing the quotient by 2 until it equals 0. The remainders are the binary number.


Example 5:


22 ÷ 2 11 ÷ 2 5 ÷ 2 2 ÷ 2 1 ÷ 2


Convert 2210 to binary =


= = = =


Example 6: Addition (i) Add 1010 + 1010 in base 2 1010 1010 10100


Multiply (ii) Multiply 11 × 11 in base 2 (iii) Multiply 101 × 10 in base 2


PROJECT 1.1


Quotient Remainder 11 . . . . . . . . . 0 5 . . . . . . . . . 1 2 . . . . . . . . . 1 1 . . . . . . . . . 0 0 . . . . . . . . . 1


Adding, multiplying and other functions are the same as base 10.


Add the columns from right to left. 0 + 0 = 0 1 + 1 = 10 Write the 0 and carry the 1 1 + 0 + 0 = 1 1 + 1 = 10


(ii) 11


× 11 11


110 1001


Base 4 Write out the first 20 numbers in base 4 (using only digits


0, 1, 2, 3). Then add 134 and 224 and multiply 24 by 134. Check your answers by converting back to base 10.


Hexadecimal arithmetic is done using base 16. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. 410 =416 1410 =E16


2410 =1816 3410 =2216


13


(iii) 101 × 10 1010


2210 = 101102


4410 =2C16


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