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B) 3 5. 65 ð 10 4. 1 Binary numbers Problem 2. The system of numbers in everyday use is the denary or decimal system of numbers, using the digits 0 to 9. It has ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and is said to have a radix or base of 10. The binary system of numbers has a radix of 2 and uses only the digits 0 and 1. e. 510 , the suffixes 2 and 10 denoting binary and denary systems of numbers respectively. Problem 1. Convert 110112 to a denary number. 0625 D1ð Problem 3. 01012 to a denary number.

This can be reduced by cancelling as 8pq in arithmetic. 2p 2ðp 1 Thus: D D 8pq 8ðpðq 4q 2b C 4b aC b 2p ł 8pq means 4 7 xy C y 2 (1) x into x 3 goes x 2 . Put x 2 x3 C 0 C 0 C y 3 above x 3 (2) x 2 x C y D x 3 C x 2 y x3 C x2 y x2 y x2 y (3) Subtract C y 3 (4) x into xy 2 (5) xy 2 C y 3 (6) xy 2 C y 3 (7) 0 0 (8) (9) x 2 y goes xy. Put xy above dividend xy x C y D x 2 y xy 2 Subtract x into xy 2 goes y 2 . Put y 2 above dividend y 2 x C y D xy 2 C y 3 Subtract 38 Basic Engineering Mathematics x3 + y3 = x 2 − xy + y 2 x +y The zeros shown in the dividend are not normally shown, but are included to clarify the subtraction process and to keep similar terms in their respective columns.

Simplify 2 ab a1/2 b3 It is usual to express the answer in the same form as the question. Hence p p p 6 a7/2 b7/6 c11/2 D a 7 b7 c 11 When a D 14 , b D 64 and c D 1, p p p 6 a7 b7 c11 D D 1 7 4 1 7 2 2 p 6 7 647 p 111 1 D1 Problem 23. Simplify a3 b a 4 b answer with positive indices only. 2 , expressing the Using the first law of indices gives: a3C 4 b1C 2 The HCF of each of the three terms is a1/2 b. Dividing each term by a1/2 b gives: Using the fifth law of indices gives: a 1 b 1 1 D aC1 bC1 ab a2 b a 3= 2 a2 b a1/2 b D D a1/2 b3 ab2 a1/2 b3 ab2 a 1= 2 b − b 2 1/2 1/2 a b a b d2 e2 f1/2 d3/2 ef5/2 answer with positive indices only.

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