MathematicsChapter 2: Logarithms
Characteristic and Mantissa
For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
Learning objectives
- Explain Characteristic And Mantissa in simple mathematical language.
- Read the comparison table and choose the correct rule or method.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
What this grid covers
For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1). If x = d x 10^n in scientific notation, the characteristic is n. The number of digits in the whole-number part of a number >= 1 is characteristic + 1. A bar over an integer, such as bar 4, means characteristic -4 with a still-positive mantissa. Never write bar 2.0899 as -2.0899, because that would make the mantissa negative.
Where / symbols
- characteristic
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- integer part
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- mantissa
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- 0 <= m < 1
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- d x 10^n
- If x = d x 10^n in scientific notation, the characteristic is n.
- characteristic is n
- If x = d x 10^n in scientific notation, the characteristic is n.
- characteristic + 1
- The number of digits in the whole-number part of a number >= 1 is characteristic + 1.
- bar 4
- A bar over an integer, such as bar 4, means characteristic -4 with a still-positive mantissa.
- -4
- A bar over an integer, such as bar 4, means characteristic -4 with a still-positive mantissa.
- bar 2.0899
- A key idea used when studying Characteristic And Mantissa.
Relationship
- 0 <= m < 1
WHAT — Core idea
For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1). If x = d x 10^n in scientific notation, the characteristic is n. The number of digits in the whole-number part of a number >= 1 is characteristic + 1. A bar over an integer, such as bar 4, means characteristic -4 with a still-positive mantissa. Never write bar 2.0899 as -2.0899, because that would make the mantissa negative.
WHY — Why it matters
This idea helps you recognise the right mathematical pattern, choose a safe method, and explain your working clearly.
HOW — Method and examples
720000: 7.2 x 105 — 5 (digits in 720000: 6 = 5+1) 4.08: 4.08 x 100 — 0 0.00072: 7.2 x 10-4 — bar 4 (that is -4) 32.5: 3.25 x 101 — 1 (2 digits = 1+1) 0.25: 2.5 x 10-1 — bar 1 1000: 1 x 103 — 3
HOW — Answer in an exam
bar 2 means characteristic -2 with mantissa still between 0 and 1. Writing bar 2.0899 as -2.0899 is wrong: -2.0899 would be characteristic -3 plus a positive mantissa 0.9101, or it would look as if the mantissa were negative. Keep the bar only on the integer part.
Read n from scientific notation; mantissa stays non-negative:
| Number x | Form d x 10^n | Characteristic |
|---|---|---|
| 1. 720000 | 7.2 x 10^5 | 5 (digits in 720000: 6 = 5+1) |
| 2. 4.08 | 4.08 x 10^0 | 0 |
| 3. 0.00072 | 7.2 x 10^-4 | bar 4 (that is -4) |
| 4. 32.5 | 3.25 x 10^1 | 1 (2 digits = 1+1) |
| 5. 0.25 | 2.5 x 10^-1 | bar 1 |
| 6. 1000 | 1 x 10^3 | 3 |
In simple words
bar 2 means characteristic -2 with mantissa still between 0 and 1. Writing bar 2.0899 as -2.0899 is wrong: -2.0899 would be characteristic -3 plus a positive mantissa 0.9101, or it would look as if the mantissa were negative. Keep the bar only on the integer part.
Key points
- characteristic
- integer part
- mantissa
- 0 <= m < 1
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Characteristic and Mantissa
Sticky Notes · Quick Revision
WHAT
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
Key Points
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- Read n from scientific notation; mantissa stays non-negative: Review each row and apply its rule carefully.
- bar 2 means characteristic -2 with mantissa still between 0 and 1.
HOW
- 720000: 7.2 x 105 — 5 (digits in 720000: 6 = 5+1)
- 4.08: 4.08 x 10^0 — 0
- 0.00072: 7.2 x 10^-4 — bar 4 (that is -4)
- 32.5: 3.25 x 101 — 1 (2 digits = 1+1)
- 0.25: 2.5 x 10^-1 — bar 1
Examples
- 720000: 7.2 x 10^5
- 4.08: 4.08 x 10^0
- 0.00072: 7.2 x 10^-4
- 32.5: 3.25 x 10^1
- 0.25: 2.5 x 10^-1
Common Mistake
- bar 2 means characteristic -2 with mantissa still between 0 and 1.
- Writing bar 2.
Exam Tip
- bar 2 means characteristic -2 with mantissa still between 0 and 1. Writing bar 2.0899 as -2.0899 is wrong: -2.0899 would be characteristic -3 plus a positive mantissa 0.9101, or it would look as if the mantissa were negative. Keep the bar only on the integer part.
60-second revision
- For a common log, split the value into characteristic (the integer part) and mantissa (the fractional part with 0 <= m < 1).
- Read n from scientific notation; mantissa stays non-negative: Review each row and apply its rule carefully.
- bar 2 means characteristic -2 with mantissa still between 0 and 1.
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