Number

Standard Form

Year 9

✓Convert very large and very small numbers into standard form and vice versa.
✓Understand the conditions for a number to be written in standard form (a x 10^n, where 1 ≤ a < 10 and n is an integer).
✓Perform multiplication and division calculations with numbers expressed in standard form.
✓Perform addition and subtraction calculations with numbers expressed in standard form.

Key concepts

Standard Form Definition

Standard form (also known as scientific notation) is a way of writing very large or very small numbers concisely. A number is in standard form when it is written as a x 10^n, where 'a' is a number such that 1 ≤ a < 10, and 'n' is an integer (a whole number, positive, negative, or zero).

a x 10^n

Large Numbers in Standard Form

For very large numbers (greater than or equal to 10), the exponent 'n' will be a positive integer. The value of 'n' tells you how many places the decimal point has moved to the left from its original position to get the 'a' value.

Small Numbers in Standard Form

For very small numbers (between 0 and 1), the exponent 'n' will be a negative integer. The absolute value of 'n' tells you how many places the decimal point has moved to the right from its original position to get the 'a' value.

Multiplying Numbers in Standard Form

To multiply numbers in standard form, multiply the 'a' values together and add the exponents ('n' values). You may need to adjust the result to ensure the 'a' part is still between 1 and 10.

(a x 10^n) x (b x 10^m) = (a x b) x 10^(n+m)

Dividing Numbers in Standard Form

To divide numbers in standard form, divide the 'a' values and subtract the exponents ('n' values). You may need to adjust the result to ensure the 'a' part is still between 1 and 10.

(a x 10^n) / (b x 10^m) = (a / b) x 10^(n-m)

Adding and Subtracting Numbers in Standard Form

To add or subtract numbers in standard form, you must first convert them so that they have the same power of 10, or convert them to ordinary numbers. Then, add or subtract the 'a' values. Finally, convert the result back to standard form if necessary.

Key facts to remember

1Standard form is written as a x 10^n.
2The value of 'a' must satisfy 1 ≤ a < 10.
3The value of 'n' must be an integer (positive, negative, or zero).
4A positive 'n' indicates a large number (greater than or equal to 10).
5A negative 'n' indicates a small number (between 0 and 1).
6When multiplying, multiply the 'a' values and add the exponents.
7When dividing, divide the 'a' values and subtract the exponents.
8When adding or subtracting, the powers of 10 must be the same before combining the 'a' values.

Worked examples

Example 1

a) Write 73 500 000 in standard form. b) Write 4.1 x 10^-5 as an ordinary number.

Ia) To get 'a' between 1 and 10, the decimal point needs to be after the 7. So, a = 7.35.

IIThe decimal point moved 7 places to the left (from the end of 73 500 000 to after the 7). So, n = 7.

IIIb) The exponent is -5, which means the number is small. Move the decimal point 5 places to the left.

IVStarting with 4.1, moving 1 place gives 0.41, 2 places gives 0.041, 3 places gives 0.0041, 4 places gives 0.00041, 5 places gives 0.000041.

Answer

a) 7.35 x 10^7. b) 0.000041

Remember that a positive exponent means a large number, and a negative exponent means a small number (between 0 and 1).

Example 2

Calculate (3 x 10^4) x (5 x 10^6). Give your answer in standard form.

IMultiply the 'a' values: 3 x 5 = 15.

IIAdd the exponents: 4 + 6 = 10.

IIIThis gives 15 x 10^10.

IVThe 'a' value (15) is not between 1 and 10. Adjust it: 15 = 1.5 x 10^1.

VSubstitute this back: (1.5 x 10^1) x 10^10 = 1.5 x 10^(1+10).

Answer

1.5 x 10^11

Always check that your final 'a' value is between 1 and 10.

Example 3

Calculate 3.2 x 10^5 + 4.5 x 10^4. Give your answer in standard form.

IMake the powers of 10 the same. It's often easier to convert to the larger power or to ordinary numbers. Let's convert 4.5 x 10^4 to a power of 10^5.

IITo change 10^4 to 10^5, we multiply by 10. To keep the number the same, we must divide the 'a' value by 10. So, 4.5 x 10^4 = 0.45 x 10^5.

IIINow add the 'a' values: 3.2 x 10^5 + 0.45 x 10^5 = (3.2 + 0.45) x 10^5.

IV3.2 + 0.45 = 3.65.

VThe result is 3.65 x 10^5.

VICheck if 3.65 is between 1 and 10. Yes, it is.

Answer

3.65 x 10^5

Alternatively, convert both to ordinary numbers: 320 000 + 45 000 = 365 000. Then convert 365 000 to standard form: 3.65 x 10^5.

Common mistakes

✗Writing 'a' outside the range 1 ≤ a < 10 (e.g., 12 x 10^3 or 0.5 x 10^7).
✗Incorrectly determining the sign of the exponent 'n' (e.g., writing 0.0006 as 6 x 10^4 instead of 6 x 10^-4).
✗Errors in counting decimal places when converting between standard form and ordinary numbers.
✗Forgetting to adjust the 'a' value after multiplication or division if it falls outside the 1 ≤ a < 10 range.
✗Attempting to add or subtract numbers in standard form without first making the powers of 10 the same or converting to ordinary numbers.

Exam tips

★Always double-check that your final answer for 'a' is between 1 and 10.
★Pay close attention to the sign of the exponent 'n' – it determines if the number is large or small.
★For addition and subtraction, if you find it difficult to adjust the powers of 10, convert both numbers to ordinary form, perform the calculation, and then convert the result back to standard form.
★Use your calculator to check answers, but be prepared to show full working for calculations in standard form, especially for non-calculator papers or questions requiring specific steps.

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