Egypt Secondary 3 Communication Technology, Week 1: Name, base and symbols: describing any number system

Digits, numbers and the decimal system

Week 1 · Unit 1 Number systems (1): digit and number, the decimal system, place values and powers of the base · 3 lessons

Name, base and symbols: describing any number system

ICT Secondary 3 Term 1, Unit 1, Topic 1: pp. 5–10

GoalDescribe a number system by its name, base and symbols.

New words

number system · a way of writing numbers with a fixed set of symbols and place valuesbase · the number of different symbols of a system, and also how many times bigger each place is than the one on its rightsymbols · the digits a system may use, for example 0 to 9subscript · a small number written low after a number to show its base

Explanation

Every number system can be described by three things: its name, its base and its symbols. The decimal system has base 10 because it has ten symbols, 0 to 9. When we count past the last symbol we write 0 in that place and add 1 to the place on its left: after 9 comes 10, which means one ten and no ones. The base also gives the place values: 1, then the base, then the base times itself, and so on. For base 10 they are 1, 10, 100, 1000. Computers use two other systems a lot: binary (base 2, symbols 0 and 1) and hexadecimal (base 16, symbols 0 to 9 and A to F). We meet them in the next weeks. Because a written number such as 110 could be read in different systems, we show the base with a small number after it: (4083)₁₀ is read in base ten.

Name	Base	Symbols	Place values from the right
decimal	10	0 to 9	1, 10, 100, 1000
binary	2	0 and 1	1, 2, 4, 8
hexadecimal	16	0 to 9 and A to F	1, 16, 256, 4096

Worked examples

Reading	Left wheel	Right wheel
08	0	8
09	0	9
10	1	0
11	1	1

A counting wheel, like the ones on an old meter, shows the symbols 0 to 9. When the right wheel passes 9 it goes back to 0 and pushes the wheel on its left one step. That is why 09 is followed by 10.

Ordinary count	Written with two symbols
0	0
1	1
2	10
3	11
4	100

Now imagine wheels with only two symbols, 0 and 1. The rule is the same: after 1 the wheel goes back to 0 and pushes the next wheel. Counting gives 0, 1, 10, 11, 100. Here is what these mean as ordinary counting numbers. A system with two symbols has base 2.

Class activity

On paper, write the name, base and symbols of the decimal system. Then invent a system with base 3 and symbols 0, 1 and 2: write its first place values (1, 3, 9) and count from the ordinary number 0 up to 6 in it.

Practice

Give the name, base and symbols of the decimal system.

A system has base 7. How many symbols does it have, and which is the last one?

Write the first four place values of a system with base 2.

Why do we write (4083)₁₀ and not just 4083 when several systems are in use?