Number Base Conversions

People count in twos, fives, twenties etc. Also the days of the week can be counted in 24 hours. Generally people count in tens. The digits 0,1,2,3,4,5,6,7,8,9 are used to represent numbers. The place value of the digits is shown in the number. Example: 395:- 3 Hundreds, 9 Tens and 5 Units. i.e.

                 395₁₀      = 3 x10² + 9 x 10¹ +5 x 10⁰.

Since the above number is based on the powers of ten, it is called the base ten number system i.e.

                               = 300 + 90 + 5.

 Also 4075 = 4 Thousand 0 Hundred 7 Tens 5 Units i.e. 4 x 10³ + 0 x 10² + 7 x 10¹ + 5 x 10⁰ Other Number systems are sometimes used.

Example: The base 8 system is based on the power of 8. For example: Expand 647₇, 26523₇, 101101₂,

(a)        645₇       = 6 x 7² + 4 x 7¹ + 5 X 7⁰ = 6 x 49 + 4 x 7 + 5 x 1

(b)        26523₇   = 2 x 7⁴ + 6 x7³ + 5 x 7² + 2 x 7¹ + 3 x 7⁰

(c)        101101₂ = 1 x 2⁵ + 0 x 2⁴ + 1 x 2³ + 1 x 2² + 0 x2¹ + 1 x 2⁰

EVALUATION

Expand The Following

1.         735₈     2.         1010011₂

CONVERSION TO DENARY SCALE (BASE TEN)

When converting from other bases to base ten the number must be raised to the base and added.

Worked Examples:

Convert the following to base 10

(a)        27₈       (b)        11011₂

Solutions:

(a)        27₈ = 2 x 8¹ + 7 x 8⁰ = 2 x 8 + 7 x 1 = 16 + 7 = 23

(b)        11011₂ = 1 x 2⁴ + 1 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰ = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1

                         = 16 + 8 + 0 + 2 + 1 = 27

EVALUATION

Convert The Following To Base Ten:

(a)        101011₂                       (b) 2120₃

CONVERSION FROM BASE TEN TO OTHER BASES

To change a number from base ten to another base

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

2.         Continue dividing until zero is reached;

3.         Write down the remainder each time;

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

Watch the video to learn base to base 10 and other bases.

EVALUATION                                                                        

1.         Convert 568₁₀ to base 8                                                 

2.         Convert 100₁₀ to base 2

Converting Binary to Decimal

EVALUATION

Convert the following bicimals to base ten.

1. 10.0001
2. 10.01
3. 11.1
4. 0.001

Conversion of number from one base to another base

A number given in one base other than base ten can be converted to another base via base ten.

Example 1

Convert:           (a) 1534_six to base eight

                        (b) 8A9F_sixteen to base eight.

EVALUATION

1. If x represents a base number in the following equations, what is the value of x?
   1. 315_x – 223_x = 72_x
      1. 405_x + 43_eight = 184_ten
   1. Convert each of the following to the base indicated:
2. 10401.11_seven to base eight
3. 4B3F_sixteen to base twelve

GENERAL EVALUATION

1. Convert

(a) 1785₁₀ to base 7        (b) Convert 2125₆ to base 10

• Determine the number bases x and y in the following simultaneous equations:
• 31_x + 20_y = 23₁₀

23_x – 11_y = 5₁₀

• 26_x – 34_y = 1000₂

38_x – 21_y = 111₅

• Find the value of Q if (Q₄)² ₌ 100100₂

READING ASSIGNMENT

New Gen Math SS 1pg52 – 51

WEEKEND ASSIGNMENT

1.         Express 342₆ as number in base 10         (a) 134        (b) 341        (c) 143

2.         Change the number 10010₂ to base 10    (a) 1001      (b) 40          (c) 18

3.         Express in base 2, 100₁₀                          (a) 100100  (b) 1100100 (c) 11001

4.         Convert 120 base 10 to base 3                (a) 11110₃    (b) 1210₃      (c) 12110₃

5.         Convert 25 base 10 to base 2                  (a) 11001₂   (b) 1001₂      (c) 1100₂

THEORY

1.         Convert 2364₇ to base 10

2.         Convert 105₁₀ to base 2