Concept of number base system Conversion from one base to base 10 Conversion of decimal fraction in one base to base 10

WEEK 2:

DATE……………………….

TOPIC: NUMBER BASE SYSTEM

CONTENT:

• Concept of number base system
• Conversion from one base to base 10
• Conversion of decimal fraction in one base to base 10

SUB-TOPIC 1:

CONCEPT OF NUMBER BASE SYSTEM. 

The number system is a way to represent or express numbers. You have heard of various types of number systems such as the whole numbers and the real numbers.  The number system that you use the most is the decimal number system. In this system, the digits 0 through 9 are used to represent numbers. The value of a digit in a number is determined by its position or place in the number. For example, the digit 7 in the number 375 has a value of 7 ones, or just 7. This is because it is in the one’s place. However, the digit 5 in that same number has a value of 5 tens, or 50. This is because it is in the ten’s place. Likewise, the digit 3 has a value of 3 hundreds, or 300.

 

A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. A base is also a number that, when raised to a particular power (that is, when multiplied by itself a particular number of times, as in 102 = 10 x 10 = 100), has a logarithm equal to the power.

For example, the logarithm of 100 to the base 10 is 2.

CONVERSION FROM ONE BASE TO BASE 10

Converting from one base to another is actually quite simple, as long as you understand the concept of place value. In order to convert from one base to another, all you need to do is determine the place values of each digit in the number and multiply each digit by its corresponding place value. The sum of all of these products will give you the number in base 10.

For example, let’s convert the number “1011” from binary to base 10. In binary, each digit corresponds to a power of 2, with the rightmost digit being worth 1, the second digit being worth 2, the third digit being worth 4, and the fourth digit being worth 8.

EVALUATION

1) What is the base 10 equivalent of the binary number 1011?

2) How do you convert from one base to another?

3) What are the place values of each digit in a binary number?

4) How do you determine the place value of a digit in a number?

5) What is the sum of the place values of a binary number?

Answer

1) The base 10 equivalent of the binary number 1011 is 11.

2) To convert from one base to another, you need to determine the place values of each digit in the number and multiply each digit by its corresponding place value.

3) The place values of each digit in a binary number are 1, 2, 4, and 8.

4) To determine the place value of a digit in a number, you need to multiply the digit by its corresponding place value.

5) The sum of the place values of a binary number is 15.

CONVERSION FROM ONE BASE TO BASE 10

Two digits—0, 1—suffice to represent a number in the binary system; 6 digits—0, 1, 2, 3, 4, 5—are needed to represent a number in the sexagesimal system; and 12 digits—0, 1, 2, 3, 4, 5 ,6, 7, 8, 9, t (ten), e (eleven)—are needed to represent a number in the duodecimal system.

The number 30155 in the sexagesimal system is the number (3 × 64) + (0 × 63) + (1 × 62) + (5 × 61) + (5 × 60) = 3959 in the decimal system; the number 2et in the duodecimal system is the number (2 × 122) + (11 × 121) + (10 × 120) = 430 in the decimal system.

To convert from any base to base ten, expand the given number(s) in the powers of their bases and simplify.

Examples:
1. Convert 1243five to base ten

Convert 1243five to base ten

The answer is 1243.5. To convert a number with a decimal point to base ten, move the decimal point to the right until there are no more digits to the left of it. Count the number of places you moved the decimal point; this is the exponent on 10 that you will use in your answer. In this case, you moved the decimal point 4 places to the right. So the answer is 1243.5 = 1.2435 × 10^4.

2. Convert 1111110two to a number in base ten.

Convert 1111110two to a number in base ten.

To convert 1111110two to a number in base ten, divide the number into groups of three digits starting from the right. In each group, multiply each digit by 2 raised to the power of its position in the group. Then, add all of the products together. This will give you the number in base ten.

For example, to convert 1111110two to a number in base ten, you would divide it into groups of three digits:

1 111 110

Then, you would multiply each digit by 2 raised to the power of its position in the group:

1*2^6 + 1*2^5 + 1*2^4 + 1*2^3 + 1*2^2 + 1*2^1 + 0*2^0

= 64+32+16+8+4+2+0

= 126

Therefore, 1111110two in base two is equal to 126 in base ten.

 

Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.

Conversion from other base less than ten to base ten

Conversion from other base greater than ten to base ten

Conversion of decimal fraction in one base to base ten

Conversion of fractions in base ten to any base

SUB-TOPIC 1:

CONVERSION FROM OTHER BASE LESS THAN TEN TO BASE TEN

Firstly, we shall consider conversion of numbers in a base less than ten to a base ten number.

A number in base ten is known as a decimal or denary number.

All numbers in a given n can be written using only the following digits 0, 1, 2 , ….., n -1.

For instance in base two, the only digits that can be used are only 0 and 1. In base three, you can only use digits 0, 1 or 2.

Generally our normal counting is done in base ten when doing this, the base is normally indicated. E.g in the denary number 546 is 546ten.

The first digit from the right towards left is 6 and is called the unit digit.

The next digit is 4 and is called the tens digit and has value 4×10.

The next digit 5 is called the hundred digit i.e 500. Hence 52 41 60 = 5 4 6 = 5×102+4×101+6×100
= 500 + 400+ 6
= 546

Notice that the digits listed on top of the denary numbers 546 are the powers of the base.

Example
Convert 1203 in base five to a denary number.

Solution
Circle the digits of the number 1203 from the last to the first, beginning at zero i.e203five

Therefore expand number raising the base five to the grades listed on the top of the number as shown below
i.e 2 0 3 five= 1×53+2×52+0x51+3×50
= 1×125+2×25+0x5+3×1
= 125+50+0+3
= 178
The new number is in base ten i.e 1203five= 178ten

This expansion method can be used in converting from any base to base ten.

Evaluation

1. What is the denary equivalent of 10.2three?

2. What is the denary equivalent of 214seven?

3. What is the denary equivalent of 780nine?

4. How do you convert a mixed number to denary?

5. How do you convert a fraction to denary?

Convert the following to denary numbers
(i) 10.2three
(ii) 214seven
(iii) 780nine

Solve for x and y if 32x + 53y + 61nine 24x + 35y = 45ten

SUB-TOPIC 2:

CONVERSION FROM OTHER BASE GREATER THAN TEN TO BASE TEN

Expansion method can be used to convert numbers in base say base thirteen to base ten, Remember in base thirteen the digits we have are 0, 1, 2, 3,4,5, 6,7,8,9, A, B, C. where A represents ten B represents eleven and C represents twelve. Letters are used for two- digits numbers less than the base thirteen.

Example
Convert 1B9thirteen to denary number Solution
1B9thirteen= 1×132+Bx131+9×130
= 1×169+11×13+9×1
= 169+143+9
= 321ten
Example: Convert 206fifteen to a denary number
Solution
20Cfifteen= 2×152+0x151+12×150
= 2×225+0x15+12×1
= 450+0x15+12×1
= 462ten

Evaluation

Convert the following to denary numbers

1. 1024eleven
2. 2059twelve
3. 51Cfourteen

ASSIGNMENT

A. Convert the following numbers to denary numbers
(i) 10011two
(ii) 768nine
(iii) 10Aeleven
(iv) B12twelve
(v) 7B3Atwelve
(vi) 6D4Fsixteen

SUB-TOPIC 3:

CONVERSION OF DECIMAL FRACTIONS IN ONE BASE TO BASE TEN

Sometimes we are faced with numbers which are not whole numbers. Hence it is very necessary to study also the conversion of fractional parts of numbers. The following examples can be used in our study of the conversion of fractional parts of other bases to decimal system.

Example 1:

Convert to denary number

1. What is the denary value of Convert to denary n= (2^6)+ (2^4) +1?

2. How do you convert a binary number to a denary number?

3. What is the difference between a binary number and a denary number?

4. What is the value of 2^6 in binary?

5. What is the value of 2^4 in binary number

Solution

Example 2:

Convert to base ten.
Solution

Example 3:
Convert to base ten. Solution

EVALUATION

1. Convert to a given number in base ten.
2. Convert the binary number to base 10.

 

SUB-TOPIC 4:

Hexadecimal Number System (Base 16)

In this system, 16 digits used to represent a given number. Thus it is also known as the base 16 number system. Each digit position represents a power of 16. As the base is greater than 10, the number system is supplemented by letters. Following are the hexadecimal symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

To take A, B, C, D, E, and F as part of the number system is conventional and has no logical or deductive reason. The letters A to F can be used in upper or lower case.

The position of the digit in a number determines its value. For example, looking at the number AB, we see that A is in the “tens” position and B is in the “ones” position. The value of A is therefore 10 (1×16) and the value of B is 1 (1×1). The value of AB is therefore 10+1 or 11. In general, the value of a digit n in position p is given by

n×16p.

The number system has a base of 16, which means that there are 16 different digits that can be used to represent numbers. The 16 digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

The value of a digit in a number is determined by its position in the number. For example:

The digit A in the number 10A has a value of 10 (1×16)

The digit B in the number AB has a value of 1 (1×1)

So the value of 10A is 10+1 or 11 and the value of AB is 10+1 or 11.

In general, the value of a digit n in position p is given by n×16p.

Examples:

Convert the following decimal numbers to hexadecimal:

a) 25

b) 42

c) 137

a) 25 = 19 in hexadecimal

b) 42 = 2A in hexadecimal

c) 137 = 89 in hexadecimal

Convert the following hexadecimal numbers to decimal:

a) 19

b) 2A

c) 89

a) 19 = 25 in decimal

b) 2A = 42 in decimal

c) 89 = 137 in decimal

The hexadecimal number system is used in computer science because it is a convenient way of representing binary numbers. Each hexadecimal digit represents four binary digits (bits). For example, the binary number 1001 can be represented as 9 inhexadecimal:

1001 = 1×24 + 0×23 + 0×22 + 1×21

= 16+0+0+1

= 17

Similarly, the binary number 1010 can be represented as A in hexadecimal:

1010 = 1×24 + 0×23 + 1×22 + 0×21

= 16+0+4+0

= 20

The number system has a base of 16, which means that there are 16 different digits that can be used to represent numbers. The 16 digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B (11), C (12), D (

Interactive Questions and Answers

Questions

1. What is the base 16 number system?

2. What are the 16 digits in the hexadecimal number system?

3. How does the position of a digit in a number determine its value?

4. What is the value of a digit n in position p in the hexadecimal number system?

5. What is the hexadecimal representation of the binary number 1001?

1. The base 16 number system is a number system that uses 16 digits to represent a given number.

2. The 16 digits in the hexadecimal number system are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B (11), C (12), D (13), E (14), and F (15).

3. The position of a digit in a number determines its value in the hexadecimal number system by n×16p, where n is the digit and p is the position.

4. For example, the value of the digit A in the number 10A would be 10 (1×16) and the value of the digit B in the number AB would be 1 (1×1).

5. The hexadecimal representation of the binary number 1001 would be 9.

 

 

EVALUATION
1. Express to base two
2. Express to decimal base.

3. Convert the following numbers in denary to the base indicated. to base 6
(b) to base 3.
4. Find the value of in each of the following equations.

(a)
(b)

(c)
(d)
Find the value of and in the following pairs of equation.

(a)

(b)

(c)

(d)

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