Numbers in Standard Form Revision Notes for Leaving Cert Mathematics

Numbers in Standard Form

What is standard form?

Standard form (also called scientific notation) is a convenient way of writing very large and very small numbers. When you use a calculator to multiply large numbers like 60,000 × 4,600,000, the screen displays $2.76 × 10^{11}$ . This means 2.76 multiplied by 10 eleven times.

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Definition: A number in the form $a × 10^n$ , where $1 ≤ a < 10$ and $n$ is an integer, is said to be expressed in standard form.

The number has two parts:

Coefficient (a): A number between 1 and 10 (including 1 but not 10)
Power ( $10^n$ ): Ten raised to an integer power

Converting large numbers to standard form

When working with large numbers, you need to move the decimal point to create a coefficient between 1 and 10, then use a positive power of 10.

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Key rule: If you move the decimal point to the left, you multiply by a positive power of 10.

Examples of large numbers:

$4000 = 4 × 1000 = 4 × 10^3$
$64,000 = 6.4 × 10,000 = 6.4 × 10^4$
$254,000 = 2.54 × 100,000 = 2.54 × 10^5$

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Pattern to remember:

1 place to the left → multiply by $10^1$
2 places to the left → multiply by $10^2$
3 places to the left → multiply by $10^3$

Converting small numbers to standard form

For numbers less than 1, you move the decimal point to the right and use negative powers of 10.

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Key rule: If you move the decimal point to the right, you multiply by a negative power of 10.

Examples of small numbers:

$0.04 = \frac{4}{100} = \frac{4}{10^2} = 4 × 10^{-2}$
$0.007 = \frac{7}{1000} = \frac{7}{10^3} = 7 × 10^{-3}$
$0.068 = \frac{6.8}{100} = \frac{6.8}{10^2} = 6.8 × 10^{-2}$

Using calculators with standard form

Most calculators have an EXP or 10ˣ key for entering numbers in standard form.

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To enter $2.54 × 10^3$ :

Press 2.54
Press the EXP or $10^x$ key
Press 3
Press equals

The calculator will display 2450 (the decimal equivalent).

Operations with numbers in standard form

Adding and subtracting

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Method: Convert each number to decimal form first, then perform the operation.

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Worked Example: Addition in Standard Form

Calculate: $2.4 × 10^2 + 1.68 × 10^3$

Solution: $2.4 × 10^2 + 1.68 × 10^3$ $= 2.4 × 100 + 1.68 × 1000$ $= 240 + 1680 = 1920 = 1.92 × 10^3$

Multiplying and dividing

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Method: First multiply/divide the coefficients, then multiply/divide the powers of 10.

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Worked Example: Multiplication in Standard Form

Calculate: $(3.8 × 10^3) × (9.4 × 10^{-2})$

Solution: $(3.8 × 10^3) × (9.4 × 10^{-2}) = (3.8 × 9.4) × 10^3 × 10^{-2}$ $= 35.72 × 10^1$ $= 3.572 × 10^2$

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Remember: When multiplying powers of 10, add the exponents. When dividing, subtract the exponents.

Worked examples

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Worked Example 1: Subtraction

Express $2.76 × 10^3 - 5.9 × 10^2$ in standard form.

Solution: $2.76 × 10^3 - 5.9 × 10^2 = 2760 - 590 = 2170 = 2.17 × 10^3$

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Worked Example 2: Complex Division

Express $(6 × 10^4) × (4.5 × 10^3) ÷ (1.2 × 10^4)$ in standard form.

Solution: $= \frac{(6 × 4.5 × 10^4 × 10^3)}{(1.2 × 10^4)}$ $= \frac{(27 × 10^7)}{(1.2 × 10^4)}$ $= (27 ÷ 1.2) × (10^7 ÷ 10^4)$ $= 22.5 × 10^3 = 2.25 × 10^4$

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Worked Example 3: Converting to Decimal

Convert $6.8 × 10^{-2}$ to decimal form.

Solution: $6.8 × 10^{-2} = 6.8 × \frac{1}{100} = \frac{6.8}{100} = 0.068$

Real-world applications

Standard form is particularly useful for expressing very large or very small quantities in science.

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Real-world Examples:

Earth's diameter: $1.27 × 10^4$ km
Distance from Earth to Sun: approximately $1.5 × 10^8$ km
Mass of an electron: $9.11 × 10^{-31}$ kg

Exam tips

Here are some essential strategies for working with standard form in exams:

Always check that your coefficient is between 1 and 10
Remember the direction rule: left movement = positive power, right movement = negative power
Use your calculator for complex operations, but show your working
Convert to decimal for addition and subtraction
Work with coefficients and powers separately for multiplication and division
Check your final answer makes sense in context

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Key Points to Remember:

Standard form expresses numbers as $a × 10^n$ where $1 ≤ a < 10$ and $n$ is an integer
Large numbers use positive powers of 10, small numbers use negative powers
For addition/subtraction: convert to decimal form first
For multiplication/division: work with coefficients and powers separately
Always use your calculator to check complex calculations
The EXP or 10ˣ key on calculators makes entering standard form numbers easier

Numbers in Standard Form (Leaving Cert Mathematics): Revision Notes