Standard form

Welcome to Standard Form!

Ever wondered how scientists talk about the distance to the Sun or the size of a tiny atom without getting a headache from all those zeros? That is where Standard Form (also called Scientific Notation) comes to the rescue! In this chapter, you will learn how to write very large and very small numbers in a neat, professional way. Don't worry if it seems a bit "maths-heavy" at first—once you see the pattern, it’s like a secret code you can easily crack.

What exactly is Standard Form?

Standard form is a way of writing any number as a value between 1 and 10, multiplied by a power of 10.
It always follows this exact recipe:
\(A \times 10^n\)

There are two golden rules for this recipe:
1. The Number (\(A\)): This must be at least 1 but less than 10. (It can be 1.5, 9.99, or 4, but never 0.5 or 12).
2. The Power (\(n\)): This must be an integer (a whole number). If the original number is huge, the power is positive. If the number is tiny (starts with 0.00...), the power is negative.

Key Takeaway:

If your first number isn't between 1 and 10, it’s not in standard form!

Converting Ordinary Numbers to Standard Form

Think of the power of 10 as a set of instructions telling you how many places to move the decimal point.

1. Large Numbers (Positive Powers)

Let’s look at the syllabus example: 1320.
Step 1: Where would the decimal point go to make this number between 1 and 10? Between the 1 and the 3! So, we get 1.32.
Step 2: How many places did we move the decimal from the end of the number to get there?
1320.0 \(\rightarrow\) 132.0 \(\rightarrow\) 13.2 \(\rightarrow\) 1.32 (That’s 3 jumps).
Step 3: Write it together: \(1.32 \times 10^3\).

2. Small Numbers (Negative Powers)

Let’s look at the syllabus example: 0.00943.
Step 1: Move the decimal point to make a number between 1 and 10. It needs to go after the 9. So, we get 9.43.
Step 2: Count the jumps from the original decimal point to the new spot.
0.00943 \(\rightarrow\) 0.0943 \(\rightarrow\) 0.943 \(\rightarrow\) 9.43 (That’s 3 jumps).
Step 3: Because the original number was tiny, the power is negative: \(9.43 \times 10^{-3}\).

Memory Aid: The "LARS" Trick

Left Add, Right Subtract.
If you move the decimal point to the Left, the power goes up (Add).
If you move the decimal point to the Right, the power goes down (Subtract).

Converting back to Ordinary Numbers

This is just the reverse!
Example: Write \(4.5 \times 10^4\) as an ordinary number.
The power is 4, so move the decimal 4 places to the right.
4.5 \(\rightarrow\) 45 \(\rightarrow\) 450 \(\rightarrow\) 4500 \(\rightarrow\) 45,000.

Example: Write \(6.2 \times 10^{-2}\) as an ordinary number.
The power is -2, so move the decimal 2 places to the left.
6.2 \(\rightarrow\) 0.62 \(\rightarrow\) 0.062.

Quick Review:

• Positive power = Big number (move decimal right).
• Negative power = Small number (move decimal left).

Ordering Numbers in Standard Form

When you need to put numbers in order of size, look at the powers first!
• A higher power of 10 always means a larger number.
• If the powers are the same, compare the first numbers (\(A\)).

Example: Which is larger: \(2 \times 10^5\) or \(9 \times 10^4\)?
Even though 9 is bigger than 2, the power \(10^5\) is bigger than \(10^4\). So, \(2 \times 10^5\) is much larger!

Calculations Without a Calculator

This is where we use the Laws of Indices you learned earlier in the section!

Multiplication

Rule: Multiply the numbers, Add the powers.
Example: \((2 \times 10^3) \times (3 \times 10^4)\)
1. Multiply the numbers: \(2 \times 3 = 6\).
2. Add the powers: \(3 + 4 = 7\).
3. Final Answer: \(6 \times 10^7\).

Division

Rule: Divide the numbers, Subtract the powers.
Example: \((8 \times 10^6) \div (2 \times 10^2)\)
1. Divide the numbers: \(8 \div 2 = 4\).
2. Subtract the powers: \(6 - 2 = 4\).
3. Final Answer: \(4 \times 10^4\).

Addition and Subtraction

This is the trickiest bit! To add or subtract, the powers must be the same.
If they aren't, convert them into ordinary numbers first, do the sum, and then convert back to standard form.

Example: \((3 \times 10^3) + (2 \times 10^2)\)
1. Convert to ordinary: \(3000 + 200 = 3200\).
2. Convert back: \(3.2 \times 10^3\).

Common Mistake to Avoid:

Students often forget to check if their final answer is still in standard form. For example, if you get \(12 \times 10^5\), you must change it to \(1.2 \times 10^6\) to get full marks!

Using a Calculator

Most modern calculators have a special button for standard form. Look for a button that says \(x10^x\), EXP, or EE.
To enter \(5 \times 10^6\), you would type: [5] [\(x10^x\)] [6].
Top Tip: Always use brackets around your numbers when typing them into the calculator to avoid order-of-operation errors!

Final Summary Checklist

• Is my number between 1 and 10?
• Is it multiplied by 10 to a power?
• For large numbers, is the power positive?
• For small numbers, is the power negative?
• When multiplying, did I add the powers?
• When dividing, did I subtract the powers?

You've got this! Standard form is just a way to make messy numbers look clean. Keep practicing those decimal jumps!