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CIE IGCSE Mathematics (0580) Standard form Study Notes

CIE IGCSE Mathematics (0580) Standard form Study Notes - New Syllabus

CIE IGCSE Mathematics (0580) Standard form Study Notes

LEARNING OBJECTIVE

  • Standard Form (Scientific Notation)

Key Concepts: 

  • Standard Form and its conversion

CIE iGCSE Maths (0580)  Study Notes – All topics

Standard Form (Scientific Notation)

Standard Form (Scientific Notation)

Definition:

Standard form is a way of writing very large or very small numbers using powers of 10. It is written as:

\( A \times 10^n \), where:

  • \( 1 \leq A < 10 \)
  • \( n \in \mathbb{Z} \) (i.e., an integer)

Why use standard form?

It makes writing and reading very large or very small numbers easier and clearer, especially in science and engineering.

Examples of Large Numbers:

  • \( 3{,}600{,}000 = 3.6 \times 10^6 \)
  • \( 5{,}000 = 5.0 \times 10^3 \)

Examples of Small Numbers:

  • \( 0.00042 = 4.2 \times 10^{-4} \)
  • \( 0.007 = 7.0 \times 10^{-3} \)

How to Convert a Number to Standard Form:

  1. Move the decimal point to get a number between 1 and 10.
  2. Count how many places you moved the decimal — this gives the power \( n \).
  3. If the original number was large (greater than 10), \( n \) is positive. If it was small (less than 1), \( n \) is negative.

Important Notes:

  • The value of \( A \) must always be at least 1 and less than 10.
  • The power of 10 tells you how many places to move the decimal point.

Example:

The mass of a red blood cell is approximately \( 3.2 \times 10^{-14} \) grams. There are about \( 5 \times 10^9 \) red blood cells in one millilitre of blood. What is the total mass of red blood cells in 1 millilitre of blood?

▶️ Answer/Explanation

Step 1: Use multiplication of standard form:

\( (3.2 \times 10^{-14}) \times (5 \times 10^9) \)

Step 2: Multiply the numbers: \( 3.2 \times 5 = 16 \)

Step 3: Add the powers of 10: \( 10^{-14} \times 10^9 = 10^{-5} \)

So, the result is \( 16 \times 10^{-5} \)

Step 4: Convert to standard form:

\( 16 = 1.6 \times 10^1 \), so: \( 16 \times 10^{-5} = (1.6 \times 10^1) \times 10^{-5} = 1.6 \times 10^{-4} \)

Final Answer: \( 1.6 \times 10^{-4} \) grams

Converting Numbers Into and Out of Standard Form

Converting Numbers Into and Out of Standard Form

1. Converting an Ordinary Number to Standard Form

To write a number in the form \( A \times 10^n \), where \( 1 \leq A < 10 \), follow these steps:

  1. Move the decimal point so that only one non-zero digit is to the left of it.
  2. Count how many places you moved the decimal point. This becomes the power \( n \).
  3. If the original number is greater than 1, \( n \) is positive. If the original number is less than 1, \( n \) is negative.

Examples:

  • \( 6700000 = 6.7 \times 10^6 \)
  • \( 0.00053 = 5.3 \times 10^{-4} \)

2. Converting From Standard Form to an Ordinary Number

To convert a number written as \( A \times 10^n \) back to ordinary form:

  1. Look at the power of 10, which tells you how many places to move the decimal.
  2. If \( n \) is positive, move the decimal to the right.
  3. If \( n \) is negative, move the decimal to the left.

Examples:

  • \( 3.2 \times 10^5 = 320000 \)
  • \( 4.5 \times 10^{-3} = 0.0045 \)

Tips:

  • Always write \( A \) as a number between 1 and 10.
  • Use zeros as placeholders when shifting the decimal point.

Example:

Write 67,000,000 in standard form.

▶️ Answer/Explanation

Step 1: Move the decimal after the first digit: \( 6.7 \)

Step 2: Count places moved: 7 digits to the right

Final Answer: \( 6.7 \times 10^7 \)

Example:

Write 0.00042 in standard form.

▶️ Answer/Explanation

Step 1: Move decimal to make it \( 4.2 \)

Step 2: Count digits moved: 4 to the right

Step 3: Use negative exponent: \( 4.2 \times 10^{-4} \)

Final Answer: \( 4.2 \times 10^{-4} \)

Example:

Write \( 7.3 \times 10^{-3} \) as an ordinary number.

▶️ Answer/Explanation

Step 1: Move decimal 3 places left: \( 0.0073 \)

Final Answer: \( 0.0073 \)

Calculations with Standard Form

Calculations with Standard Form

Multiplying Numbers in Standard Form

To multiply two numbers in standard form: \( (A \times 10^m) \times (B \times 10^n) = (A \times B) \times 10^{m+n} \)

Steps:

  1. Multiply the numbers \( A \) and \( B \).
  2. Add the powers of 10.
  3. Adjust the result if the new \( A \) is not in the range \( 1 \leq A < 10 \).

Dividing Numbers in Standard Form

To divide two numbers in standard form: \( \frac{A \times 10^m}{B \times 10^n} = \left( \frac{A}{B} \right) \times 10^{m – n} \)

Steps:

  1. Divide the numbers \( A \div B \).
  2. Subtract the powers of 10.
  3. Adjust the result if necessary to keep \( A \) within \( 1 \leq A < 10 \).

Adding and Subtracting in Standard Form

You can only add or subtract numbers in standard form **if their powers of 10 are the same**.

Steps:

  1. Rewrite the numbers so that both powers of 10 match.
  2. Add or subtract the decimal parts.
  3. Write the result in standard form.

Important Tips:

  • After any calculation, always check that the number part \( A \) is between 1 and 10.
  • Round answers where required, usually to 2 or 3 significant figures.

Example:

Multiply: \( (3.2 \times 10^4) \times (2.5 \times 10^3) \)

▶️ Answer/Explanation

Step 1: Multiply the numbers: \( 3.2 \times 2.5 = 8.0 \)

Step 2: Add the exponents: \( 10^4 \times 10^3 = 10^{4+3} = 10^7 \)

Final Answer: \( 8.0 \times 10^7 \)

Example:

Divide: \( \frac{4.5 \times 10^6}{1.5 \times 10^2} \)

▶️ Answer/Explanation

Step 1: Divide the numbers: \( \frac{4.5}{1.5} = 3 \)

Step 2: Subtract the powers: \( 10^6 \div 10^2 = 10^{6 – 2} = 10^4 \)

Final Answer: \( 3 \times 10^4 \)

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