Writing Numbers in Standard Form

Grade 4

What is standard form?

Standard form is a way of writing very large or very small numbers more neatly. Instead of writing 4 500 000 or 0.000 045, you express them as a number between 1 and 10, multiplied by a power of 10.

a × 10ⁿ where 1 ≤ a < 10 and n is an integer

The key rule: a must be at least 1 but less than 10. That means there's exactly one non-zero digit before the decimal point. If your number doesn't satisfy that, you haven't finished yet.

Converting large numbers

For a large number, you're moving the decimal point to the left until only one digit sits before it. Each step left increases the power of 10 by one.

Count how many places you move the decimal point — that count becomes your positive power of 10.

EXAMPLE

Write 34 700 000 in standard form.

Move the decimal point 7 places to the left to get 3.47. So the number is 3.47 × 10⁷. Check: 3.47 is between 1 and 10 ✓. The power is positive because the original number is large. 3.47 × 10⁷

Converting small numbers

For a small number (a decimal less than 1), move the decimal point to the right until one non-zero digit sits before it. This time the power of 10 is negative — it shows how small the number is.

EXAMPLE

Write 0.000 062 in standard form.

Move the decimal point 5 places to the right to get 6.2. The power is −5 because the number is small. 6.2 × 10⁻⁵. Check: 6.2 is between 1 and 10 ✓, and 10⁻⁵ = 0.00001, so 6.2 × 0.00001 = 0.000062 ✓

Converting back from standard form

To reverse the process, look at the power. A positive power means move the decimal right (making the number bigger). A negative power means move it left (making it smaller).

The most common mistake is getting the direction wrong when converting back. Positive power → big number, negative power → small number. If your answer looks the wrong size, you've gone the wrong way.

Try this one

Put these numbers in order of size, starting with the smallest.

\[ 5.1 \times 10^3, \quad 4.8 \times 10^4, \quad 9.2 \times 10^2, \quad 6.3 \times 10^3 \]

[2]

Nova's hint:

This uses the technique of ordering numbers given in standard form by comparing their powers first, then their values of \( a \).

Ask yourself: 'Which numbers have the same power of 10, and how do I compare those?' Try converting each to an ordinary number first if it helps.

Try to Solve it with Nova?