Whole Numbers 1

Mathematics J.S.S 3 First Term

Theme: Number and Numeration

Sub Theme: Whole Number

WEEK 1

Whole Numbers 1

Performance Objectives

Students should be able to;

1. Use computer to do simple mathematical calculations
2. Simplify Word problems

Number Base System

Content

Number Base is a system of counting natural numbers in bundles. Some languages have their own unique method or way of counting numbers while others have same method. For instance, numbers are counted in bundle of ten digits called base ten or denary.  The digits involved in English natural numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Most people count in tens. For instance, the place value of the digits in number 4956 in this system

Thousands     hundreds          tens        units

↓                       ↓                   ↓             ↓

4                      9                   5             6

The place values are in powers of ten. It is called the base ten system.

4956 = 4 X 1000 + 9 X 100 + 5 X 10 + 6 X 1

= 4  X  10³  + 9 X 10²  + 5 X 10¹ + 6 X 1⁰

Some unique number bases include: Base 2 ( Binary), Base 8 (Octal), Base 10 (Denary or Decimal), Base 16 (Hexadecimal),… there are other bases such as base 3, base 4, base 5, base 6, etc. The bases of number systems are usually written as a subscript either in word or figure. Examples: 111011₂ or 111011_two, 5734₈ or 5734_eight, 781₁₀ or 781_ten, etc.

Some people traditionally count in 5s, other in 20s. Using the method of the base ten system, a base five system is in powers of five. So, for example, in base five, 342 would be

342_five = 3 X 25 + 4 X 5 + 2 X 1

  = 3 X 5²  + 4 X 5¹+ 2 X 5⁰

Notice that 342_five is short for 342 in base five.

NOTE;

Converting base ten numbers to other bases

1. To convert from base ten to another base, express the given number in powers of the new base.
2. Numbers in base ten can be written without the base indicated. Therefore any number base that is written without base indicated should be considered as denary or base ten numbers.

The Binary Number System

Binary numbers are a system of counting numbers in base two. In a binary system, the greatest digit is 1 and the least or lowest is 0. Hence, the two digits available in binary numbers are 1 and 0. Binary numbers are the most important number bases because of its usage in computer. Examples of binary numbers are 11111₂, 1000001_two, 0101₂, etc.

 

Expressing Binary as Sum of Multiples

We can easily express any binary number as power of multiples of two as shown in the following examples;

1. 10001_tw0  = (1 × 2⁴) + (0 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)
2. 1111₂ = (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)
3. 110_two = (1 × 2²) + (1 × 2¹) + (0 × 2⁰)

Conversion of Binary Numbers to Decimal Numbers

To express binary numbers in decimals or denary, we write the binary number as a sum of multiples of powers of two, or we multiply each digit by the base and add to the next digit starting from the left.

NOTE: Any number or letter raised to the power of zero is 1

Examples;

1.  Convert 1111_two to denary.

Solution

1111_two = (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)

    =     8      +       4     +      2     +      1

    = 15_ten

2.  Express 11101_two as a decimal number

Solution

11101_two = (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)

     =     16     +      8      +      4     +      0      +     1

     = 29_ten

3.  Convert 1111110_two to a number in base ten.

Solution

1111110_two = (1 × 2⁶) + (1 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰)

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

= 126_ten

Writing Numbers in Other Base or Conversion From Denary Numbers to Any Other Base

In converting a denary number to base two (binary) or any other base, we divide by that new base keeping the remainder in each step until there is nothing more to divide. The result is the list of remainders from the last to the first.

Examples:                    

1. Convert 41_ten to base five
2. Convert 243_ten to base eight
3. Express 26₁₀ as a binary number

Solutions:

a. ∴ 41_ten = 131_five

b. ∴ 243_ten = 363_eight

c. ∴ 26_ten = 11010_two

 Or

Convert 37_ten  a. to base eight,  b. to base five.

a. Since 37 < 64, there are no sixty-fours in 37. To find the number of eights in 37, divide by 8.

37 ÷ 8 = 4, remainder 5

37 = 4 eights + 5 units

37_ten = 45_eight

Check:  45_eight = 4 X 8 + 5 X 1

= 32 + 5

= 37

Since 37 > 25, there must be a twenty-five in 37.

37 ÷ 25 = 1, remainder 12

37 = 1 twenty - five + 12 units.

Consider the 12 units. Since 12 > 5, there must be some fives in 12.

12 ÷ 5 = 2, remainder 2

12 = 2 fives + 2 units

∴ 37 = 1 twenty-five + 2 fives + 2 units

= 1 x 5² + 2 X 5¹ + 2 X 1

37_ten = 122_five

Check: 122_five = 1 X 25 + 2 X 5 + 2 X 1

= 25 + 10 + 2

= 37

The method in part b of Example 2 can be shortened as follows:

5 | 37

5 |   7 + 2 (i.e  7 X 5 + 2 X 1)

5 |  1 + 2  ( i.e. 1 X 5² + 2 X 5¹)

5 |  0 + 1  ( i.e. 0 X 5³ + 1 X 5 ²)

↑                           ↑

Continued division by 5 gives remainders. Reading the remainders upwards gives    37_ten = 122_five

(see the arrows above).

To change from base ten to another base:

1. Divide the base ten number by the new base number.

2. Continue dividing until 0 (zero) is reached, writing down the remainder each time.

3. Start at the last remainder and read upwards to get the answer.

Converting in Other Bases to Base Ten, Base Two and Other Bases

To do this, two steps are involved:

Step 1: Convert the given base to base ten.

Step 2: Convert the result gotten from step1 above to the required base.

Example:

1. Convert 3041_five to base four.

Solution:

First convert 3041_five to base 10

3041_five = (3 × 5³) + (0 × 5²) + (4 × 5¹) + (1 × 5⁰)

= 375 + 0 + 20 + 1 = 395_ten

Now, convert 395_ten  to base 4.

∴ 3041_five = 395_ten = 12023_four

2. Convert 10111_two to base 6.

Solution:

Step 1: We first convert 10111_two to base ten.

10111_two = (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)

= 16+ 0 + 4 + 2 + 1 = 23_ten

Step 2: We now convert 23_ten to base 6.

∴ 10111_two = 23_ten = 35_six

 

Convert an Octal into a Decimal
This is a little bit of the same as before except our base is now 8 instead of 6. 

1. Convert 723₈ to base ten

Solution
723₈ = (7 X 8²) + (2 X 8¹) + (3 X 8⁰)
       =   448     +     16     +    3
= 467₁₀

723 octal is 467 decimal.  See, that’s not so tough.  Now, the astute reader may have noticed another way this can be accomplished using the same information from the original formula.