Notes — ShuleDirect

Mathematics

BASE

Jump to topic

BASE

In this chapter, students will learn the concept of bases in numeration systems. They will explore how to count objects in different bases, read and write base numerations, and understand their meaning. They will also practice changing numbers between bases and performing arithmetic operations within and across bases.

Concept of base

A base (or radix) is the number of unique digits used to represent numbers in a number system.Examples:

Base 10 → Digits: 0–9
Base 2 → Digits: 0, 1
Base 3 → Digits: 0, 1, 2
Base 5 → Digits: 0, 1, 2, 3, 4

Use base in the classroom

Activity 1

Study the following examples. Objects can be grouped according to a chosen base.

Base 10 grouping, take 23 counters (23 = 2 groups of ten + 3 ones)
Base 5 grouping, take 17 counters (17 = 3 groups of five + 2 ones)
Base 2 grouping, take 6 counters (6 = 3 groups of two + 0 ones)
Base 3 grouping, take 14 counters (14 = 4 groups of three + 2 ones)
Base 5 grouping, take 25 counters (25 = 5 groups of five + 0 ones)
Base 2 grouping, take 9 counters (9 = 4 groups of two + 1 one)
Base 3 grouping, take 10 counters (10 = 3 groups of three + 1 one)
Base 10 grouping, take 45 counters (45 = 4 tens + 5 ones)
Base 5 grouping, take 8 counters (8 = 1 five + 3 ones)
Base 2 grouping, take 7 counters (7 = 3 twos + 1 ones)

Exercise 1

Group the following in the indicated base:

18 objects in base 5
11 objects in base 3
9 objects in base 2
36 objects in base 10
14 objects in base 5
8 objects in base 3
5 objects in base 2
62 objects in base 10
22 objects in base 5
13 objects in base 3

Count objects in base (10, 5, 3 and 2)

Activity 2

Observe the following examples. Counting follows place values of the base.

Exercise 2

Write the next number

Base 2: 101, ___
Base 3: 12, ___
Base 5: 34, ___
Base 2: 111, ___
Base 3: 22, ___
Base 5: 44, ___
Base 2: 10, ___
Base 3: 20 , ___
Base 5: 14, ___
Base 2: 110, ___

Read and write base numerations

Activity 3

Reading a Base 5 Number: Observe the following examples

243₅ Read as: Two-four-three base five.
204₅ Read as:Two-zero-four base five
432₅Read as: Four-three-two base five.
1001₅ Read as: One-zero-zero-one base five.
3402₅ Read as: Three-four-zero-two base five.

Activity 4

Reading a base 3 number" Study the following examples

210₃ Read as Two-one-zero base three.
2002₃Read as Two zero zero two base three
1201₃Read as One two zero one base three
22021 Read as Two two zero two one base three

Activity 5

Reading a base 2 number: Look the following examples

1011₂Read as: One-zero-one-one base two.
1110₂Read as: One-one -one-zero base two
1002 Read as: One-zero-zero-two base two

Exercise 3

Write the following numbers in words

324₅
2. 1222₃
3. 10110₂
4. 22102₃
5. 4432₅

Exercise 4

Write the following numbers in numerals

Four three two base five
Two one two zero base three
One zero zero zero base two
One two one zero base three
One four three zero base five

Explain the meaning of base numerations

Activity 6

Read the following information

Base numeration (or number bases) refers to the system of counting and representing numbers using a fixed set of digits.
The "base" tells us how many unique digits are available before we need to "carry over" to the next place value.
Key Points
Definition: The base of a number system is the number of digits it uses to represent numbers.
Place Value: Each position in a number represents a power of the base.
Base 10 (Decimal): Uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Base 5 (Quinary): Uses digits 0, 1, 2, 3 and 4
Base 3 ( Ternary): Uses digits 0, 1 and 2
Base 2 (Binary): Uses digits 0 and 1