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CHAPTER 1: NUMBER BASES.
Introduction [1]
A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0.
The most commonly used number system is the decimal system, commonly known as base 10. In everyday life, we count or estimate quantities using groups of ten items or units. This may be so because, naturally, we have ten ngers. For example, when we count ten, i.e. we write 10 meaning one group of 10 and no units. A quantity like twenty ve, written as 25 means 2 groups of 10 and 5 units
Suppose instead we had say 6 fingers;
How, in your opinion would we do our counting?
If we had eight fingers, how would we count?
This is now what we are to cover under this topic.
NOTE:
The digits of a number in any base are less than the base itself.
The digits 10 and 11 are represented by t and e respectively in number bases.
For digits above 11 are represented by alphabetic letters of your choice.
The names of some number systems is as given below.
Activity: Getting familiar with number bases:
Bases are used in day today life. Therefore copy and complete the table below by giving some real life situations were bases are used.
1.1 Identifying numbers of different bases on an abacus
1. Which possible base does each abacus below represent.
(b) Write down the numbers represented on the abaci above.
2. Write down the numbers represented on the abaci below.
Activity: List the numerals for the following bases.
Numerals are digits(or symbols) that are used for writing numbers in a given base. The digits are always less than the base itself. study the table below and fill in the gaps.
1.2 Place Values Using the Abacus
The representation of numbers on an abacus helps in identifying the place value of digits in any base
Activity: Making abaci
1. Make abaci for the following number bases.
2. Make an abacus for any number in base twelve.
Activity : Reading and stating the value of digits in bases.
1. State the place value and value of digit for each numeral in the following numbers.
1.1 Exercise Set
1. State the place value of each numeral in the following numbers:
2. State the value of each numeral in the following numbers
1.3 Converting Numbers
Numbers can be converted from one base to another, and when you do this, you get the same numbers written in different bases.
1.3.1 Converting from any base to base ten
EXAMPLES
Convert the following to base ten.
1.2 Exercise Set.
1.3.2 Converting from base ten to other bases.
We use BNR.
Divide the number repeatedly by the required bases.
The remainder in reverse order gives the required number.
1.3 Exercise Set.
1.4 Operation on Numbers in Various Bases.
In this section we are going to look at the four mathematical operations which include addition, subtraction, division and multiplication.
Activity: James had two jackfruit trees in his compound. At one time one tree had 8 fruits ready and the other 7 fruits. He harvested them at the same time.
If James puts the jack fruits in heaps of ten fruits. How many heaps of ten did he get and how many remained?
If James puts the jack fruits in heaps of nine fruits. How many heaps of nine did he get and how many remained?
If James puts the jack fruits in heaps of five fruits. How many heaps of five did he get and how many remained?
When you put the fruits in heaps of 10,9 and 5, you are adding in base 10,base 9 and base 5.
1.4.1 Addition of bases
If the sum of the digits exceeds the base, divide that sum by the base then write down the remainder and carry the whole number.
EXAMPLES
1.4 Exercise Set.
1.4.2 Subtraction of bases
In case of borrowing the new value is the sum of the base and the digit which was small
EXAMPLES
1.4.3 Multiplication of bases
Find the product of any two numbers as we do in base ten
Divide this product by the base number
Write the remainder and carry the quotient to the next place value position.
EXAMPLES
1.6 Exercise Set .
1. Fill in the missing numbers in this multiplication table in base twelve
2. Workout the following leaving your answer in the base indicated
1.4.4 Division of bases
Convert each number base to base ten.
Divide the two numbers in base ten.
Convert the result back to the required base.
EXAMPLES
1.7 Exercise Set
Activity of intergration
On April 4, 2020 the Covid19 task force started the distribution of food in Ayivu Division( Arua district). Each member in the household was given a package containing 6 kgs of maize our, and 3 kg of beans. There are 10 households in the community with 3, 5, 7, 4, 6, 5,8,12, 13,4 members respectively.
TASK
1. Determine the number of packages the task force distributed in Ayivu division.
2. Determine the total weight of the maize our that was distributed in the division.
3. In case there are some remaining packages, discuss what the task force should do with them.
4. The prices of, beans and maize our was approximated to be at 4000UGX and 2500UGX per kilogram respectively. What is the total amount of money spent by the government on maize our and beans in the 10 households?
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