Cambridge International Mathematics

2 Investigation and modelling questions (Chapter 34)

23

Look at this table of numbers. Row 11

Row23 5

Row37 9 11

Row413 15 17 19

...... ...... ...... ...... ...... ......

...... ...... ...... ...... ...... ...... ......

aiWrite down the sum of each of the first four rows.

ii What is the sum of the numbers in the hundredth row? Explain your answer.

biWrite down the sum of all the numbers in the first two rows

in the first three rows

in the first four rows

in the firstnrows.

ii What is the sum of all the numbers in the first 100 rows?

c What is the last number in thenth row?

24

a Convert these binary numbers to decimal numbers:

i 1012 ii 10012 iii 11 111 2 iv 1 001 001 2 v 11 011 011 2

b Convert these decimal numbers into binary numbers:

i 7 ii 17 iii 39 iv 81 v 138 vi 432

c Thehexadecimal systemuses base 16 and contains the digits 0 to 9 as well as the letters A to F.

This is also used in computers as hexadecimal numbers are very easily converted to binary ones.

Notice that BED 16 =11£ 162 +14£ 161 +13£1 = 3053 10

i Convert the following hexadecimal numbers into decimal numbers:

a 9816 b 21316 c DAD 16 d 1 BC (^416)
ii Convert these decimal numbers into hexadecimal numbers:
a 123 b 389 c 2164 d 18 341
iii Convert these hexadecimal numbers into binary numbers:
a 2516 b BE 16 c 1 A 416 d ABE 16
In ourdecimal systemthe number 2358 represents 2 thousands, three hundreds, five tens and 8 ones.
That is, 2358 = 2000 + 300 + 50 + 8 or 2 £ 103 +3£ 102 +5£ 101 +8£ 1
All positive integers can be written in this form where thebaseis 10.
Thebinary systemusesbase 2. So, place values of digits correspond to powers of 2 (instead of 10 ).
Each number contains only the digits 0 and 1. Binary numbers are used in a computer. Numbers are
stored in binary code.
For example, 101 011 2 =1£ 25 +0£ 24 +1£ 23 +0£ 22 +1£ 21 +1
So, 101 011 2 has a decimal value of 32+8+2+1=43.
To convert a decimal number to a binary number, observe the following division method:
2 43
2 21 + 1
2 10 + 1
2 5+0
2 2+1
2 1+0
0+1 So, 43 = 1£ 25 +0£ 24 +1£ 23 +0£ 22 +1£ 21 +1
Adapted from Nov 1998, Paper 4
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