Conversion from Base Ten to other Bases and Conversion from one base to another

WEEK 3

DATE……………………….

CONVERSION OF NUMBER FROM ONE BASE TO ANOTHER

Content

1. Conversion from base ten to other bases

2. Conversion from one base to another

base via base ten. We convert from base ten to other bases by repeated division

Example. Convert 137ten to a base five number

5 137

5 5 r 2

5 1 r 0
0 r 1

: . 137ten = 1022five

137ten = 2425five

Example :

Converts the denary number 102 to a base twelve number

12 107

12 8 r 11
0 r 8

107ten = 8Btwelve

To convert from a base to another you may have to pass through base ten

Example, Convert 301four to a base six number.

Solution
First 301fourwill be converted to a base ten number 301four = 3×42 + 0x41 + 1×40
= 48 + 0 + 1
= 49ten

49tenwill now be converted to a base six number by repeated division 6 49
6 8 r 1
6 1 r 2

0 r 1 301four = 121six

Evaluation

1. Convert 2210three to a base five number
2. Convert 5201seven to a binary (base two) number

Addition, Subtraction and Multiplication of number

Operation in other bases other than base ten are carried out in a manner similar to what is obtained in base ten. We can illustrate the procedure as shown in the example below.
Example. 167eight + 125eight Solution
167eight

+ 145eight

7+5 = 12. This exceeds the value of the base. 12 contain a bundle of 8 and 4 units. That one bundle of 8 is carried to the next column as 1
1 + 6 + 4 = 11
11 is another single bundle of 8 and three, Hence we write 3 and carry the bundle to the next column as 1
167eight

+ 145eight 334eight
Example 501twelve – 3Btwelve
501eight

+ 3Beight

Recall B in base twelve is eleven.
If 1 is ‘borrowed from 5 in the third column, getting to the next column on the right becomes a twelve. From it we can take one to the next column to the right again. To get 12+1 = 13 from which we finally subtract B (i.e eleven)
501twelve

+ 3Btwelve 482 twelve

Notice that after borrowing 1 from the middle column, eleven was left. If is out of this eleven that 3 is subtracted to get 8 in the second column of the answer.
Example. Simply 134six x 5six 154six

+ 5six

5 x 4 = 20 i.e 3 bundles of 6 plus 2 units. Write 2 add 3 to the product of 5 x 5 of second column to get 28. 28 = 4(sixes) plus 4. Take the 4 bundles to next column. 4 + 5x 1 = 9 which is 13six. So 154six x 5six = 1342 six
Example. Simplify 134five x 24five
134five

x 24five

4 x 4 = 16 i.e 3(fives) and 1 unit.
These 3 bundle of 5 is added to the product of (3×4) of the second column. 3×4+3 = 15. 15 = 3(fives) and zero
This new 3 bundles of 3 is to be added to the product 1x 4 of the third column
1 x 4 + 3 = 7 which will written as 12five similar thing is done with 134five times the distance 2, thus

134five

x 24five 1201
323

4431five
Evaluation
1. 1205six x 3six
2. 143five + 24five
3. 211four + 32four

4. 103four x 32four Division of numbers bases.

Since in binary (base two) system, the digits we have are 0 and 1. Each digit of the quotient 110111 ÷ 101 must be either 1 or 0.

1011
101 110111

101
11

0
111

101
101

101
Once you start the division, the digits are brought down one after the other. Example 240six÷ 20six

12
20 240

20
40

40
So 240six÷ 20six
Evaluation simplify the following
1. 4 7 7eight 2. BBtwelve 3. 1011two x 111two

+36 7eight + A1twelve

4. Which is bigger E5Asixteen or 1271fifteen
5. 387nine÷ 25nine
ASSIGNMENT
New General Mathematics SS1. Ex 3i Nos 1 – 5 page 54.
Functional Mathematics Graduated Exercise page 17 – 18, No 1- 30

In there are 16 different digits that

Addition, Subtraction and Multiplication of number in base four.

The base four number system is a positional numeral system with a base of four. In the base four number system, the numbers 0 to 3 are used to represent all possible values. The value of each position in a base four number is quadrupled, making it easy to perform addition, subtraction, and multiplication.

To add two base four numbers, simply line the numbers up and add them as you would any other base ten number. For example, to add the base four numbers 2 and 3, you would line them up like this:

2

3

—

5

To subtract two base four numbers, line the numbers up and subtract them as you would any other base ten number. For example, to subtract the base four numbers 3 and 2, you would line them up like this:

3

2

—

1

To multiply two base four numbers, simply multiply them as you would any other base ten number. For example, to multiply the base four numbers 2 and 3, you would multiply them as follows:

2

3

—

6

It is important to note that in the base four number system, there is no concept of carrying over or borrowing. So, when adding or subtracting, if a result is greater than 4, the excess is simply dropped. For example, if you were to add the base four numbers 3 and 2, the result would be 1, not 7.

Similarly, when multiplying two base four numbers, if a result is greater than 4, the excess is simply dropped. For example, if you were to multiply the base four numbers 2 and 3, the result would be 6, not 12.

The base four number system is a simple way to perform addition, subtraction, and multiplication. By understanding how to add, subtract, and multiply in base four, you can easily solve problems in base four.

Presentation

The topic is presented step by step

Step 1:

The subject teacher revises the previous topics

Step 2.

He or she introduces the new topic.

Step 3:

The subject teacher allows the pupils to give their own examples and he corrects them when the needs arise