Standard form

1. Overview

Standard form, also known as scientific notation, is a way to express very large or very small numbers in a compact and consistent format. It's written as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer. This notation simplifies calculations and comparisons, especially in scientific and engineering contexts where dealing with many zeros can be cumbersome. The power $n$ indicates how many places the decimal point must be shifted to convert the number back to its ordinary form.

Key Definitions

• Standard Form: A way of writing numbers as a value between 1 (inclusive) and 10 (exclusive) multiplied by a power of 10. It takes the form $A \times 10^n$.
• Mantissa ($A$): The numerical factor in standard form. It satisfies the condition $1 \le A < 10$.
• Exponent ($n$): The integer power of 10. It indicates the number of decimal places the decimal point has been moved. A positive $n$ indicates a large number, and a negative $n$ indicates a small number (less than 1).
• Ordinary Number: A number written in its full decimal form (e.g., 5200 instead of $5.2 \times 10^3$).

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Core Content

Writing Numbers in Standard Form

To convert an ordinary number to standard form:

1. Place the decimal point after the first non-zero digit to find $A$, ensuring $1 \le A < 10$.
2. Count how many places the decimal point has moved from its original position to its new position. This count gives you the value of $n$.
3. If the original number was greater than or equal to 10, $n$ is positive.
4. If the original number was between 0 and 1, $n$ is negative.

Worked example 1 — Converting a large number to standard form

Question: Convert 678,000 to standard form.

• Step 1: Identify the first non-zero digit: 6. Place the decimal point after it: 6.78000, which simplifies to 6.78.
  • Reason: To obtain the mantissa A, where 1 ≤ A < 10.
• Step 2: Count the number of places the decimal point has moved from the end of the original number to its new position between the 6 and 7: $6\underbrace{7}{1}\underbrace{8}{2}\underbrace{0}{3}\underbrace{0}{4}\underbrace{0}_{5}$. The decimal point moved 5 places.
  • Reason: To determine the value of the exponent n.
• Step 3: Since 678,000 is greater than 10, the exponent $n$ is positive: $n = 5$.
  • Reason: Large numbers have positive exponents in standard form.
• Step 4: Write the number in standard form: $6.78 \times 10^5$.
  • Reason: Combining the mantissa and the exponent.

Answer: $\boxed{6.78 \times 10^5}$

Worked example 2 — Converting a small number to standard form

Question: Convert 0.0000409 to standard form.

• Step 1: Identify the first non-zero digit: 4. Place the decimal point after it: 4.09.
  • Reason: To obtain the mantissa A, where 1 ≤ A < 10.
• Step 2: Count the number of places the decimal point has moved from its original position to its new position between the 4 and 0: $0.\underbrace{0}{1}\underbrace{0}{2}\underbrace{0}{3}\underbrace{0}{4}\underbrace{4}_{5}09$. The decimal point moved 5 places.
  • Reason: To determine the value of the exponent n.
• Step 3: Since 0.0000409 is less than 1, the exponent $n$ is negative: $n = -5$.
  • Reason: Small numbers have negative exponents in standard form.
• Step 4: Write the number in standard form: $4.09 \times 10^{-5}$.
  • Reason: Combining the mantissa and the exponent.

Answer: $\boxed{4.09 \times 10^{-5}}$

Converting Out of Standard Form

To convert a number from standard form ($A \times 10^n$) to an ordinary number, move the decimal point in $A$ by $n$ places.

• If $n$ is positive, move the decimal point to the right (making the number larger). Add zeros as placeholders if needed.
• If $n$ is negative, move the decimal point to the left (making the number smaller). Add zeros as placeholders if needed.

Worked example 3 — Converting from standard form to an ordinary number

Question: Convert $9.12 \times 10^{-4}$ to an ordinary number.

• Step 1: Identify the exponent: $n = -4$.
  • Reason: To determine the direction and number of places to move the decimal point.
• Step 2: Since $n$ is negative, move the decimal point 4 places to the left.
  • Reason: Negative exponents indicate small numbers.
• Step 3: Add zeros as placeholders: $0.000912$.
  • Reason: To correctly position the decimal point.

Answer: $\boxed{0.000912}$

Worked example 4 — Converting from standard form to an ordinary number

Question: Convert $2.8 \times 10^{6}$ to an ordinary number.

• Step 1: Identify the exponent: $n = 6$.
  • Reason: To determine the direction and number of places to move the decimal point.
• Step 2: Since $n$ is positive, move the decimal point 6 places to the right.
  • Reason: Positive exponents indicate large numbers.
• Step 3: Add zeros as placeholders: $2,800,000$.
  • Reason: To correctly position the decimal point.

Answer: $\boxed{2,800,000}$

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Extended Content (Extended Only)

Extended students must be proficient in performing arithmetic operations (addition, subtraction, multiplication, and division) with numbers expressed in standard form, often without relying on a calculator. This requires a solid understanding of the laws of indices and careful attention to maintaining the correct standard form format.

Multiplication and Division

When multiplying or dividing numbers in standard form, the mantissas and the powers of 10 are treated separately. The laws of indices are crucial: $10^a \times 10^b = 10^{a+b}$ and $10^a \div 10^b = 10^{a-b}$. After performing the operation, the result must be adjusted back into standard form if the mantissa is not within the range $1 \le A < 10$.

Worked example 5 — Multiplication in standard form

Question: Calculate $(3.0 \times 10^5) \times (7.0 \times 10^{-2})$. Give your answer in standard form.

• Step 1: Multiply the mantissas: $3.0 \times 7.0 = 21$.
  • Reason: Multiplying the numerical parts of the standard form numbers.
• Step 2: Multiply the powers of 10: $10^5 \times 10^{-2} = 10^{5 + (-2)} = 10^3$.
  • Reason: Applying the laws of indices for multiplication.
• Step 3: Combine the results: $21 \times 10^3$.
  • Reason: Combining the mantissa and the exponent.
• Step 4: Adjust to standard form. Since 21 is not between 1 and 10, rewrite it as $2.1 \times 10^1$.
  • Reason: Ensuring the mantissa is within the required range for standard form.
• Step 5: Substitute back: $(2.1 \times 10^1) \times 10^3 = 2.1 \times 10^{1+3} = 2.1 \times 10^4$.
  • Reason: Applying the laws of indices for multiplication.

Answer: $\boxed{2.1 \times 10^4}$

Worked example 6 — Division in standard form

Question: Calculate $(9.6 \times 10^7) \div (3.2 \times 10^2)$. Give your answer in standard form.

• Step 1: Divide the mantissas: $9.6 \div 3.2 = 3$.
  • Reason: Dividing the numerical parts of the standard form numbers.
• Step 2: Divide the powers of 10: $10^7 \div 10^2 = 10^{7 - 2} = 10^5$.
  • Reason: Applying the laws of indices for division.
• Step 3: Combine the results: $3 \times 10^5$.
  • Reason: Combining the mantissa and the exponent.

Answer: $\boxed{3 \times 10^5}$

Addition and Subtraction

Numbers in standard form can only be added or subtracted directly if they have the same power of 10. If the powers are different, one of the numbers must be adjusted so that the powers match. Alternatively, both numbers can be converted to ordinary numbers, added/subtracted, and then converted back to standard form.

Worked example 7 — Addition in standard form

Question: Calculate $(4.5 \times 10^6) + (8.2 \times 10^5)$. Give your answer in standard form.

• Step 1: Make the powers of 10 the same. Convert $8.2 \times 10^5$ to $0.82 \times 10^6$.
  • Reason: To align the decimal places for addition.
• Step 2: Add the mantissas: $4.5 + 0.82 = 5.32$.
  • Reason: Adding the numerical parts of the standard form numbers.
• Step 3: Combine the results: $5.32 \times 10^6$.
  • Reason: Combining the mantissa and the exponent.

Answer: $\boxed{5.32 \times 10^6}$

Worked example 8 — Subtraction in standard form

Question: Calculate $(6.7 \times 10^{-3}) - (5.5 \times 10^{-4})$. Give your answer in standard form.

• Step 1: Make the powers of 10 the same. Convert $5.5 \times 10^{-4}$ to $0.55 \times 10^{-3}$.
  • Reason: To align the decimal places for subtraction.
• Step 2: Subtract the mantissas: $6.7 - 0.55 = 6.15$.
  • Reason: Subtracting the numerical parts of the standard form numbers.
• Step 3: Combine the results: $6.15 \times 10^{-3}$.
  • Reason: Combining the mantissa and the exponent.

Answer: $\boxed{6.15 \times 10^{-3}}$

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Key Equations

The general form is:

$\qquad \bf{A \times 10^n}$

Where:

• $A$: The mantissa, where $1 \le A < 10$.
• $n$: The exponent, which is an integer.

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Common Mistakes to Avoid

• ❌ Mantissa Out of Range: Expressing a number as $25.6 \times 10^4$.
  • ✓ Right: The mantissa must be between 1 and 10. Corrected form: $2.56 \times 10^5$.
• ❌ Incorrect Sign on Exponent: Writing $0.0075$ as $7.5 \times 10^3$.
  • ✓ Right: Small numbers have negative exponents. Corrected form: $7.5 \times 10^{-3}$.
• ❌ Forgetting Order of Operations: When using a calculator, not using the EXP button and incorrectly calculating $3 \times 10^8 + 5 \times 10^7$ as $(3 \times 10) ^ {8+5}$.
  • ✓ Right: Use the EXP or $\times 10^x$ button to enter standard form numbers, or convert to ordinary numbers first.
• ❌ Incorrect Subtraction of Negative Exponents: Calculating $(6 \times 10^5) \div (2 \times 10^{-2})$ as $3 \times 10^3$.
  • ✓ Right: Remember to subtract a negative: $5 - (-2) = 7$, so the answer is $3 \times 10^7$.

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Exam Tips

• Calculator Proficiency: Familiarize yourself with your calculator's standard form (scientific notation) functions. The $\times 10^x$ or EXP button is your friend! Use it to avoid errors, especially in complex calculations.
• Answering the Question: Always double-check the question's requirements. If it explicitly asks for the answer in standard form, providing an ordinary number, even if correct, will result in a loss of marks.
• Significant Figures: Pay attention to significant figures. If the given values are to a certain number of significant figures, your final answer should be rounded accordingly.
• Contextual Problems: Be prepared to apply standard form in real-world scenarios. These often involve very large or very small quantities, such as astronomical distances, the size of cells, or the speed of light ($3 \times 10^8$ m/s).
• Non-Calculator Strategies: In non-calculator papers, questions are often designed with numbers that simplify easily. Look for opportunities to simplify the mantissas before dealing with the powers of 10. For example, in $6.4 \times 10^8 \div 1.6 \times 10^2$, divide 6.4 by 1.6 first.