Lesson Notes By Weeks and Term - Junior Secondary School 1
BASE NUMBERS
BASE NUMBERS
SUBJECT: MATHEMATICS
CLASS: JSS 1
DATE:
TERM: 2nd TERM
REFERENCE BOOKS
TOPIC: BASE NUMBERS Content Number Bases (Expansion of Base Numbers ) When counting days in a week, we count in 7’s, but when counting seconds in a minute, we count in 60’s. However, for most purposes, people count in 10’s. The digits 0, 1,2, 3, 4, 5,6, 7, 8, 9 are used to represent numbers. The placing of the digits shows their value . For example, 7 8 0 9 means 7809 = 7 x 1000 + 8 x 100 + 0 x 10 + 9 x 1 = 7 x 103 + 8 x 102 + 0 x 101 + 9 x 100 (Note : Any number raised to the power zero = 1) since the illustration above is based on the power of 10, It is called base 10. We can write it as 7809 ten Other number systems are sometimes used. For instance 145 eight , means 145eight= 1 x 82 + 4 x 81 + 5 x 80 = 1 x 82 + 4 x 81 + 5 x 1 Example 1 Expand the following in the powers of their bases Solution Using the model provided above = 2 x 103 + 3 x 102 + 8 x 101 + 9 x 100 = 2 x 103 + 3 x 102 + 8 x 101 + 9 x 1 = 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20 = 1 x 23 + 0 x 22 + 0 x 21 + 1 x 1 = 6 x 82 + 4 x 81 + 7 x 80 = 6 x 82 + 4 x 81 + 7 x 1 Evaluation : Expand the following base numbers in the powers of their bases. From the example above, (b) was 1001 two, this means 1001 in base two. The first thing to notice is their base two number or BINARY NUMBER, is made up of only two digits 0 and 1(just as in base ten there are ten digits: ), 1, 2, 3, 4, 5,6,7,8,9,) In summary Base two ________ 0, 1 Base three ________ 0, 1,2, Base four _________ 0, 1,2, 3. etc The place value of the digits in the binary number 1111two is as shown below: Eight (23) Fours(22) Two(21) Units(20) Class Activity Work in pairs. Get a collection of about 25 counters ( e.g. matchsticks, bottle tops, smooth pebbles) Make a paper abacus and use it to answer the following questions. You will discover that nine is made up of (d) Represent the binary number for 9 n your paper abacus. IMPORTANCE OF BINARY SYSTEM The binary system is second in importance to our usual base ten system. It is important because it is used in computer programs. Binary numbers are made up of only two digits, 1 and 0. A computer contains a large number of stitches. Each switch in either ‘on’ or ‘off’. An ‘on’ switch represents 1; and ‘off’ switch represents 0. See the table below for the first ten binary numbers Base ten number Binary number 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 III. Addition in Base Two Remember the following : 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 Example 1. Calculate in base two 1 0 1 + 1 0 1 Solution 1 0 1 + 1 0 1 1 0 1 0 Note: 1st column : 1 + 1 = 0, write down 0 carry 1 2nd column : 0 + 0 + 1 carried = 1, write down 1 carry 0 3rd column: 1 + 1 + 0 carried = 10 Example 2 Simplify the following in base two + 1 1 1 ______________ + 1 __________ + 1 1 0 _________ Solution + 1 1 1 11 1 0 0 Note: 1st column :1 + 1 = 10, write 0 carry 1 2nd column: 0 + 1 + 1 carried = 10, write 0 carry 1 3rd column: 1 + 1 + 1 carried = 11, write 1 carry 1 4th column: 0 + 1 carried = 1, write 1 carry 0 5th column: 1 + 0 carried = 1 = 11100 two Using the above explanation try out the examples worked by your teacher below: + 1 1 0 0 0 ( c) 1 0 1 + 1 1 0 1 0 1 1 Evaluation: Simplify the following in base two 1 1 0 1 + 1 0 1 ___________ 1 0 1 + 1 1 1 ___________ Note: You may also need to listen to teacher’s other approach in the class to see the one you will prefer. For instance: 4 8 9 + 3 8 2 8 7 1 This is because it is in base ten. Once, it is 10 or more than your teacher told you in addition of whole numbers that we carry. When it is less than 10 you write down the number. The same thing is happening in base two. Once it is or more you must carry when it is 2 you write down 0 carry 1. i.e 2 2 = 1 remainder 0 Usually, we write the remainder and carry the quotient. See illustration 1 1 1 + 1 1 1 0 1 1 1 + 1 = 2 ( 2/2 = 1 r0 ) 1 + 1 + 1 carried = 3 ( 3/2 = 1 r 1 ) 1 + 1 carried = 2 ( 2/2 = 1 r 0 ) the answer = 1 0 1 0 two Subtraction in Base two Example 1 Simplify in base two Solution 1 1 1 - 1 1 0 _______ 1 Ans = 1 two Example 2 Simplify in base two 1 1 1 1 0 - 1 1 0 1 1 0 0 0 1 ______________ Ans = 10001 two Example 3 Simplify in base two 1 0 0 1 1 - 1 1 0 ____________ Solution 1 0 0 1 1 - 1 1 0 1 1 0 1 Ans = 1101 two Note, the same method we used when we were subtracting whole numbers is still the method we have used. The only difference is their bases. The whole number was in base 10. e.g 4 8 3 - 2 9 6 4 8 3 - 2 9 6 1 8 7 In the above example, when the number we are to subtract is larger, we borrow from the next digit. For instance, we borrowed 1 from 8 reducing it to 7 and increasing 3 to 13. Each 1 borrowed is equal to 10 which represents the base. In our own case, any 1 borrowed is equal to two representing the base. Try your self in base eight and 1 borrowed is equal to_____ Evaluation Simplify the following in base two - 1 0 1 1 1 ________ - 1 1 1 1 _____________ - 1 1 _________ Reading Assignment i.Multiplication in base two Weekend Assignment (a) 0 and 1 (b) ) and 2 (c ) 1 and 3 (d) 0, 1 and 2 (a) 5 x 92 + 9 x 81 + 6 x 90 (b) 5 x 93 + 8 x 91 + 6 x 1 (c ) 5 x 93 + 8 x 92 + 6 x 91 (d) 5 x 92 + 8 x 91 + 6 x90 Theory Simplify the following in base two 1a 1 1 1 0 + 1 0 0 1 __________ + 1 1 1 ___________ c). 1 1 0 1 two + 1 1 0 0 1 two + 1 0 1 1 two 2a) 1 1 0 1 1 - 1 0 1 1 1 ______________ - 1 0 1 1 1 _______________
WEEK FOUR Date :……….
b) 1 0 1
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