How Do You Write a Number in Scientific Notation (a × 10^k)?

Write the number as $a \times 10^k$ by moving the decimal point until the coefficient $a$ has exactly one non-zero digit before the decimal point. The number of places moved determines the integer exponent $k$.

In IB Mathematics: Applications and Interpretation, this is called standard form and belongs to SL 1.1, Using standard form. For a positive number, the coefficient must satisfy $1 \leq a < 10$; more generally, including negative numbers, $1 \leq |a| < 10$, while $k$ is an integer.

The Method

1. Move the decimal point to produce a coefficient between $1$ and $10$ in magnitude.
2. Count how many places the decimal point moved.
3. Choose the sign of the exponent:

For example, consider $48300$. Moving the decimal point four places left gives $4.83$, so

$$48300=4.83\times 10^4.$$

For $0.00627$, move the decimal point three places right:

$$0.00627=6.27\times 10^{-3}.$$

The negative exponent appears because $10^{-3}=\frac{1}{1000}$. Therefore, $6.27\times 10^{-3}=0.00627$.

For a negative number, retain the negative sign: $-920000=-9.2\times 10^5$.

You can verify each result by reversing the multiplication. For $4.83\times 10^4$, multiplying by $10^4$ moves the decimal point four places right, returning $48300$. For $6.27\times {10}^{-}$, multiplying by moves it three places left. This reverse check is especially useful when deciding whether the exponent should be positive or negative.

Also, zero is a special case: it is normally written as $0$, rather than standard form, because no coefficient satisfying $1\leq |a|<10$ can represent it. If rounding is required, apply it to the coefficient while keeping the power of ten unchanged.

A common misconception is that the exponent is always positive. This is incorrect: numbers between $0$ and $1$ require a negative exponent.

IB Exam Technique

Check that the coefficient has only one non-zero digit before the decimal point and that your exponent is an integer. Never give calculator notation such as 4.83E4 as the final answer; IB requires $4.83\times 10^4$.