MathematicsChapter 2: Logarithms
Scientific Notation
Huge and tiny measurements are messy in ordinary decimal form.
Learning objectives
- Explain Scientific Notation in simple mathematical language.
- Apply numbers less than 1 correctly.
- Apply numbers greater than or equal to 1 correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
Huge and tiny measurements are messy in ordinary decimal form. Scientific notation writes a positive number as d x 10^n where the coefficient satisfies 1 <= d < 10 and n is an integer. The power of ten shifts the point: 0.00072 = 7.2 x 10-4, 720000 = 7.2 x 105, and a number already between 1 and 10, such as 4.08 = 4.08 x 100. This form makes size obvious and prepares the ground for logarithms.
Where / symbols
- Scientific notation
- Scientific notation writes a positive number as d x 10^n where the coefficient satisfies 1 <= d < 10 and n is an integer.
- d x 10^n
- Scientific notation writes a positive number as d x 10^n where the coefficient satisfies 1 <= d < 10 and n is an integer.
- 1 <= d < 10
- Scientific notation writes a positive number as d x 10^n where the coefficient satisfies 1 <= d < 10 and n is an integer.
- 0.00072 = 7.2 x 10^-4
- A key idea used when studying Scientific Notation.
- 720000 = 7.2 x 10^5
- A key idea used when studying Scientific Notation.
- 4.08 = 4.08 x 10^0
- A key idea used when studying Scientific Notation.
- 10^0
- 08 x 10^0.
Relationship
- 1 <= d < 10
- 0.00072 = 7.2 x 10-4
- 720000 = 7.2 x 105
- 4.08 = 4.08 x 100
Overview
Move the decimal point until the coefficient sits in [1, 10):
Numbers Less Than 1
If the number is smaller than 1, n is negative. Count how many places you move the point to the right to land in 1 <= d < 10. That count is the size of the negative exponent. So 0.00072 becomes 7.2 x 10^-4 because the point jumped four places.
Examples 0.00072 = 7.2 x 10^-4 · 0.05 = 5 x 10^-2 · 0.003 = 3 x 10^-3
Numbers Greater Than Or Equal To 1
If the number is 1 or more, n is zero or positive. Count places you move the point to the left. 720000 = 7.2 x 105 (five places). A value already in range needs n = 0: 4.08 = 4.08 x 100. Never write 72 x 10^4 as scientific notation, because 72 is not less than 10.
Examples 720000 = 7.2 x 10^5 · 4.08 = 4.08 x 10^0 · 350 = 3.5 x 10^2
WHAT — Core idea
Huge and tiny measurements are messy in ordinary decimal form. Scientific notation writes a positive number as d x 10^n where the coefficient satisfies 1 <= d < 10 and n is an integer. The power of ten shifts the point: 0.00072 = 7.2 x 10-4, 720000 = 7.2 x 105, and a number already between 1 and 10, such as 4.08 = 4.08 x 100. This form makes size obvious and prepares the ground for logarithms.
WHY — Why it matters
The integer n is the order of magnitude. Two numbers with the same n are in the same power-of-ten band. Scientific notation is unique if you truly keep 1 <= d < 10.
HOW — Method and examples
1. NUMBERS LESS THAN 1: If the number is smaller than 1, n is negative. 2. NUMBERS GREATER THAN OR EQUAL TO 1: If the number is 1 or more, n is zero or positive.
HOW — Answer in an exam
The coefficient must satisfy 1 <= d < 10. Writing 0.72 x 10^-3 or 72 x 10^3 is not standard scientific notation. 10^0 = 1, so 4.08 x 10^0 is correct, not a special exception.
Explanation
The integer n is the order of magnitude. Two numbers with the same n are in the same power-of-ten band. Scientific notation is unique if you truly keep 1 <= d < 10.
In simple words
The coefficient must satisfy 1 <= d < 10. Writing 0.72 x 10^-3 or 72 x 10^3 is not standard scientific notation. 10^0 = 1, so 4.08 x 10^0 is correct, not a special exception.
Key points
- Scientific notation
- d x 10^n
- 1 <= d < 10
- 0.00072 = 7.2 x 10-4
- negative
- 0.00072
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Scientific Notation
Sticky Notes · Quick Revision
WHAT
- Huge and tiny measurements are messy in ordinary decimal form.
Key Points
- Huge and tiny measurements are messy in ordinary decimal form.
- If the number is smaller than 1, n is negative.
- If the number is 1 or more, n is zero or positive.
- The integer n is the order of magnitude.
HOW
- 1. NUMBERS LESS THAN 1: If the number is smaller than 1, n is negative.
- 2. NUMBERS GREATER THAN OR EQUAL TO 1: If the number is 1 or more, n is zero or positive.
Examples
- 0.00072 = 7.2 x 10-4
- 0.05 = 5 x 10-2
- 0.003 = 3 x 10-3
- 720000 = 7.2 x 105
- 4.08 = 4.08 x 100
Common Mistake
- The coefficient must satisfy 1 <= d < 10.
- Writing 0.
Exam Tip
- The coefficient must satisfy 1 <= d < 10. Writing 0.72 x 10^-3 or 72 x 10^3 is not standard scientific notation. 10^0 = 1, so 4.08 x 10^0 is correct, not a special exception.
60-second revision
- Huge and tiny measurements are messy in ordinary decimal form.
- If the number is smaller than 1, n is negative.
- If the number is 1 or more, n is zero or positive.
- The integer n is the order of magnitude.
- The coefficient must satisfy 1 <= d < 10.
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