Number Bases | Jamb Mathematics

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"Preparing for an exam is like solving a number base conversion problem—it requires understanding the system, practicing the process, and double-checking your results. Just as you would carefully convert numbers from base 10 to base 2, approach your study with precision, step by step, ensuring you grasp each concept thoroughly. Remember, consistency in preparation will yield results as predictable as the outcome of a well-calculated arithmetic operation."

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Number Base? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Number Base together and move one step closer to achieving your exam success! Blissful learning.

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Jamb Lesson notes on operations in different number bases from 2 to 10

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Addition

Add 1011 and 1101 in base 2.
- Step 1: Align numbers and add bit by bit:
```
    1011
  + 1101
  ------
   11000 (carry applied as needed)
```
- Final result: $11000_2$ .
Add 245 and 135 in base 6.
- Step 1: Convert to base 10:
  $245_6 = 89_{10}, 135_6 = 59_{10}$ .
- Step 2: Add in base 10: $89 + 59 = 148$ .
- Step 3: Convert back to base 6: $148 = 404_6$ .
Add 2134 and 1242 in base 5.
- Step 1: Add each digit, applying carry:
```
    2134
  + 1242
  ------
    3431
```
- Final result: $3431_5$ .
Add 342 and 123 in base 8.
- Step 1: Digit-by-digit addition in base 8:
```
    342
  + 123
  ------
    465
```
- Final result: $465_8$ .
Add 321 and 112 in base 4.
- Step 1: Add each digit, applying carry:
```
    321
  + 112
  ------
    1033
```
- Final result: $1033_4$ .

Subtraction

Subtract 11011 from 100001 in base 2.
- Step 1: Subtract bit by bit, borrowing where necessary:
```
    100001
  - 11011
    ------
    10010
```
- Final result: $10010_2$ .
Subtract 425 from 1342 in base 6.
- Step 1: Convert to base 10:
  $1342_6 = 284_{10}, 425_6 = 137_{10}$ .
- Step 2: Subtract in base 10: $284 - 137 = 147$ .
- Step 3: Convert back to base 6: $147 = 403_6$ .
Subtract 241 from 400 in base 5.
- Step 1: Subtract digit by digit in base 5:
```
    400
  - 241
  ------
    204
```
- Final result: $204_5$ .
Subtract 735 from 1346 in base 8.
- Step 1: Convert to base 10:
  $1346_8 = 750_{10}, 735_8 = 477_{10}$ .
- Step 2: Subtract in base 10: $750 - 477 = 273$ .
- Step 3: Convert back to base 8: $273 = 421_8$ .
Subtract 1002 from 2100 in base 3.
- Step 1: Perform base 3 subtraction:
```
    2100
  - 1002
  ------
    1021
```
- Final result: $1021_3$ .

Multiplication

Multiply 101 and 11 in base 2.
- Step 1: Binary multiplication:
```
    101
  x  11
  -----
    101
   1010
  -----
   1111
```
- Final result: $1111_2$ .
Multiply 123 and 32 in base 4.
- Step 1: Convert to base 10:
  $123_4 = 27_{10}, 32_4 = 14_{10}$ .
- Step 2: Multiply: $27 \times 14 = 378$ .
- Step 3: Convert back to base 4: $378 = 13222_4$ .
Multiply 241 and 32 in base 5.
- Step 1: Convert to base 10:
  $241_5 = 71_{10}, 32_5 = 17_{10}$ .
- Step 2: Multiply: $71 \times 17 = 1207$ .
- Step 3: Convert back to base 5: $1207 = 4322_5$ .
Multiply 47 and 36 in base 8.
- Step 1: Convert to base 10:
  $47_8 = 39_{10}, 36_8 = 30_{10}$ .
- Step 2: Multiply: $39 \times 30 = 1170$ .
- Step 3: Convert back to base 8: $1170 = 2202_8$ .
Multiply 12 and 21 in base 3.
- Step 1: Convert to base 10:
  $12_3 = 5_{10}, 21_3 = 7_{10}$ .
- Step 2: Multiply: $5 \times 7 = 35$ .
- Step 3: Convert back to base 3: $35 = 1102_3$ .

Division

Divide 1110 by 10 in base 2.
- Step 1: Convert to base 10:
  $1110_2 = 14_{10}, 10_2 = 2_{10}$ .
- Step 2: Divide: $14 \div 2 = 7$ .
- Step 3: Convert back to base 2: $7 = 111_2$ .
- Final result: $111_2$ .
Divide 243 by 11 in base 5.
- Step 1: Convert to base 10:
  $243_5 = 73_{10}, 11_5 = 6_{10}$ .
- Step 2: Divide: $73 \div 6 = 12{R} 1$ .
- Step 3: Convert back to base 5: $12 = 22_5, 1 = 1_5$ .
- Final result: $22 {R} 1$ .
Divide 321 by 4 in base 6.
- Step 1: Convert to base 10:
  $321_6 = 121_{10}, 4_6 = 4_{10}$ .
- Step 2: Divide: $121 \div 4 = 30 {R} 1$ .
- Step 3: Convert back to base 6: $30 = 50_6, 1 = 1_6$ .
- Final result: $50 {R} 1$ .
Divide 751 by 14 in base 8.
- Step 1: Convert to base 10:
  $751_8 = 489_{10}, 14_8 = 12_{10}$ .
- Step 2: Divide: $489 \div 12 = 40 {R} 9$ .
- Step 3: Convert back to base 8: $40 = 50_8, 9 = 11_8$ .
- Final result: $50 {R} 11$ .
Divide 213 by 10 in base 3.
- Step 1: Convert to base 10:
  $213_3 = 23_{10}, 10_3 = 3_{10}$ .
- Step 2: Divide: $23 \div 3 = 7 {R} 2$ .
- Step 3: Convert back to base 3: $7 = 21_3, 2 = 2_3$ .
- Final result: $21{R} 2$ .
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Converting number from any base to base 10

Be aware that that the dot $(\cdot)$ represent multiplication $(\times)$

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21. Convert

1011_2

to Base 10**

Step 1: Expand using powers of 2: $1011_2 = 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0$
Step 2: Calculate: $= 8 + 0 + 2 + 1 = 11$
Final result: $11_{10}$ .

Convert $245_8$ to Base 10

Step 1: Expand using powers of 8: $245_8 = 2 \cdot 8^2 + 4 \cdot 8^1 + 5 \cdot 8^0$
Step 2: Calculate: $= 128 + 32 + 5 = 165$
Final result: $165_{10}$ .

Convert $2134_5$ to Base 10

Step 1: Expand using powers of 5: $2134_5 = 2 \cdot 5^3 + 1 \cdot 5^2 + 3 \cdot 5^1 + 4 \cdot 5^0$
Step 2: Calculate: $= 250 + 25 + 15 + 4 = 294$
Final result: $294_{10}$ .

Convert $123.4_8$ to Base 10

Step 1: Expand integer part using powers of 8: $123_8 = 1 \cdot 8^2 + 2 \cdot 8^1 + 3 \cdot 8^0 = 64 + 16 + 3 = 83$
Step 2: Expand fractional part: $0.4_8 = 4 \cdot 8^{-1} = 4 \cdot 0.125 = 0.5$
Final result: $123.4_8 = 83.5_{10}$ .

Convert $110.01_2$ to Base 10

Step 1: Expand integer part: $110_2 = 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 4 + 2 + 0 = 6$
Step 2: Expand fractional part: $0.01_2 = 0 \cdot 2^{-1} + 1 \cdot 2^{-2} = 0 + 0.25 = 0.25$
Final result: $110.01_2 = 6.25_{10}$ .

Convert $7A_16$ to Base 10

Step 1: Expand using powers of 16: $7A_{16} = 7 \cdot 16^1 + 10 \cdot 16^0$
Step 2: Calculate: $= 112 + 10 = 122$
Final result: $122_{10}$ .

Convert $325.21_5$ to Base 10

Step 1: Expand integer part: $325_5 = 3 \cdot 5^2 + 2 \cdot 5^1 + 5 \cdot 5^0 = 75 + 10 + 5 = 90$
Step 2: Expand fractional part: $0.21_5 = 2 \cdot 5^{-1} + 1 \cdot 5^{-2} = 0.4 + 0.04 = 0.44$
Final result: $325.21_5 = 90.44_{10}$ .

Convert $123_4$ to Base 10

Step 1: Expand using powers of 4: $123_4 = 1 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0$
Step 2: Calculate: $= 16 + 8 + 3 = 27$
Final result: $27_{10}$ .

Convert $1A3_16$ to Base 10

Step 1: Expand using powers of 16: $1A3_{16} = 1 \cdot 16^2 + 10 \cdot 16^1 + 3 \cdot 16^0$
Step 2: Calculate: $= 256 + 160 + 3 = 419$
Final result: $419_{10}$ .

Convert $243_6$ to Base 10

Step 1: Expand using powers of 6: $243_6 = 2 \cdot 6^2 + 4 \cdot 6^1 + 3 \cdot 6^0$
Step 2: Calculate: $= 72 + 24 + 3 = 99$
Final result: $99_{10}$ .

Convert $101.11_2$ to Base 10

Step 1: Expand integer part: $101_2 = 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 4 + 0 + 1 = 5$
Step 2: Expand fractional part: $0.11_2 = 1 \cdot 2^{-1} + 1 \cdot 2^{-2} = 0.5 + 0.25 = 0.75$
Final result: $ 101.11_2 = 5.75_10 .

Convert $1001_2$ to Base 10**

Step 1: Expand: $1001_2 = 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$
Step 2: Calculate: $= 8 + 0 + 0 + 1 = 9$
Final result: $9_{10}$ .

Convert $54_7$ to Base 10

Step 1: Expand: $54_7 = 5 \cdot 7^1 + 4 \cdot 7^0$
Step 2: Calculate: $= 35 + 4 = 39$
Final result: $39_{10}$ .

Convert $1.101_2$ to Base 10**

Step 1: Expand integer part: $1_2 = 1$
Step 2: Expand fractional part: $0.101_2 = 1 \cdot 2^{-1} + 0 \cdot 2^{-2} + 1 \cdot 2^{-3} = 0.5 + 0 + 0.125 = 0.625$
Final result: $1.101_2 = 1.625_{10}$ .

Convert $321_4$ to Base 10

Step 1: Expand: $321_4 = 3 \cdot 4^2 + 2 \cdot 4^1 + 1 \cdot 4^0$
Step 2: Calculate: $= 48 + 8 + 1 = 57$
Final result: $57_{10}$ .

Convert $23.14_5$ to Base 10

Step 1: Expand integer part: $23_5 = 2 \cdot 5^1 + 3 \cdot 5^0 = 10 + 3 = 13$
Step 2: Expand fractional part: $0.14_5 = 1 \cdot 5^{-1} + 4 \cdot 5^{-2} = 0.2 + 0.16 = 0.36$
Final result: $23.14_5 = 13.36_{10}$ .

Convert $11_3$ to Base 10

Step 1: Expand: $11_3 = 1 \cdot 3^1 + 1 \cdot 3^0 = 3 + 1 = 4$
Final result: $4_{10}$ .

Convert $44_7$ to Base 10

Step 1: Expand: $44_7 = 4 \cdot 7^1 + 4 \cdot 7^0 = 28 + 4 = 32$
Final result: $32_{10}$ .

Convert $56_9$ to Base 10

Step 1: Expand: $56_9 = 5 \cdot 9^1 + 6 \cdot 9^0 = 45 + 6 = 51$
Final result: $51_{10}$ .

Convert $30.12_4$ to Base 10**

Step 1: Expand integer part: $30_4 = 3 \cdot 4^1 + 0 \cdot 4^0 = 12$
Step 2: Expand fractional part: $0.12_4 = 1 \cdot 4^{-1} + 2 \cdot 4^{-2} = 0.25 + 0.125 = 0.375$
Final result: $30.12_4 = 12.375_{10}$ .

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Thank you for taking the time to read my blog post! Your interest and engagement mean so much to me, and I hope the content provided valuable insights and sparked your curiosity. Your journey as a student is inspiring, and it’s my goal to contribute to your growth and success.

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If you found the post helpful, feel free to share it with others who might benefit. I’d also love to hear your thoughts, feedback, or questions—your input makes this space even better. Keep striving, learning, and achieving! 😊📚✨

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I recommend you check my Post on the following:

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- Jamb Mathematics- Tutorial notes on Fractions, Decimals, Approximations and Percentages for utme Success

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This is all we can take on "Jamb Mathematics - Lesson Notes on Number Bases for UTME Candidate"

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Number Bases | Jamb Mathematics

Jamb Lesson notes on operations in different number bases from 2 to 10

Addition

Subtraction

Multiplication

Division

Converting number from any base to base 10

I recommend you check my Post on the following:

POSCHOLARS TEAM.

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