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Number - Standard Form (Scientific Notation)

Grade 12IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition: Standard form (scientific notation) is written as a×10na \times 10^n, where 1≤∣a∣<101 \le |a| < 10 and nn is an integer.

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Positive Power (n>0n > 0): Represents large numbers. The decimal point moves nn places to the right from the original position.

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Negative Power (n<0n < 0): Represents small numbers between 0 and 1. The decimal point moves nn places to the left.

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Normalization: After performing calculations, if the resulting aa is not between 1 and 10, the decimal must be shifted and the exponent nn adjusted accordingly.

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Comparison: To compare numbers in standard form, first compare the powers of 10. If the powers are equal, compare the values of aa.

📐Formulae

Standard Form: a×10n(1≤a<10,n∈Z)a \times 10^n \quad (1 \le a < 10, n \in \mathbb{Z})

Multiplication: (a×10n)×(b×10m)=(a×b)×10n+m(a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}

Division: (a×10n)÷(b×10m)=(a÷b)×10n−m(a \times 10^n) \div (b \times 10^m) = (a \div b) \times 10^{n-m}

Addition/Subtraction: (a×10n)±(b×10n)=(a±b)×10n(a \times 10^n) \pm (b \times 10^n) = (a \pm b) \times 10^n (Indices must be the same before adding/subtracting)

💡Examples

Problem 1:

Convert 0.00000450.0000045 into standard form.

Solution:

4.5×10−64.5 \times 10^{-6}

Explanation:

To move the decimal point so that the first digit is between 1 and 10, we move it 6 places to the right. Since the original number is small (less than 1), the exponent is negative.

Problem 2:

Calculate (4×105)×(5×104)(4 \times 10^5) \times (5 \times 10^4), giving your answer in standard form.

Solution:

2×10102 \times 10^{10}

Explanation:

First, multiply the numbers: 4×5=204 \times 5 = 20. Next, add the powers: 105+4=10910^{5+4} = 10^9. This gives 20×10920 \times 10^9. To convert to standard form, 2020 becomes 2×1012 \times 10^1, so 2×101×109=2×10102 \times 10^1 \times 10^9 = 2 \times 10^{10}.

Problem 3:

Solve (6.3×107)+(8×106)(6.3 \times 10^7) + (8 \times 10^6).

Solution:

7.1×1077.1 \times 10^7

Explanation:

To add these, the powers of 10 must be the same. Convert 8×1068 \times 10^6 to 0.8×1070.8 \times 10^7. Now add: (6.3+0.8)×107=7.1×107(6.3 + 0.8) \times 10^7 = 7.1 \times 10^7.