Standard Form: Scientific Notation with Worked Examples

Writing Very Big and Very Small Numbers

Light travels about 300,000,000 metres every second. The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kg. Numbers like these are hard to read, easy to miscopy and tiresome to calculate with. Standard form, also called scientific notation, solves the problem by writing every number as a value between 1 and 10 multiplied by a power of 10. The speed of light becomes 3 × 10⁸ m/s, and the electron's mass becomes 9.1 × 10⁻³¹ kg. This tutorial covers standard form for BECE, WAEC, NECO and JAMB mathematics: the definition, converting large and small numbers, multiplying and dividing, adding and subtracting, rounding to significant figures, and word problems, with worked examples.

What Is Standard Form?

A number is in standard form when it is written as:

A × 10ⁿ, where 1 ≤ A < 10 and n is a whole number (an integer).

A must be at least 1 and less than 10. So 3.7 × 10⁴ is correctly written, but 37 × 10³ and 0.37 × 10⁵ are not, even though they have the same value.
n is positive for large numbers (10 or more), zero for numbers from 1 up to 10, and negative for numbers less than 1.

Standard form relies on the laws of indices: 10⁴ = 10,000, 10⁰ = 1, and 10⁻³ = 1/10³ = 0.001. The tutorial on indices and logarithms covers these laws in full.

Converting Large Numbers to Standard Form

Place the decimal point after the first non-zero digit, to get A.
Count how many places the decimal point has moved to the left. That number is n (positive).

Worked example 1

Write (a) 345,000 and (b) 7,020,000 in the form A × 10ⁿ.

(a) The point moves 5 places left: 3.45000, so 345,000 = 3.45 × 10⁵.

(b) The point moves 6 places left: 7,020,000 = 7.02 × 10⁶.

A quick check: 10⁵ = 100,000, and 3.45 × 100,000 = 345,000.

Converting Small Numbers to Standard Form

Place the decimal point after the first non-zero digit.
Count how many places the point has moved to the right. Then n is that number with a negative sign.

Worked example 2

Write (a) 0.000072 and (b) 0.000506 in the form A × 10ⁿ.

(a) The point moves 5 places right to give 7.2, so 0.000072 = 7.2 × 10⁻⁵.

(b) The point moves 4 places right to give 5.06, so 0.000506 = 5.06 × 10⁻⁴.

A useful shortcut for small numbers: n is minus the position of the first non-zero digit after the decimal point. In 0.000072, the 7 is in the 5th decimal place, so n = −5.

Converting from Standard Form to Ordinary Numbers

Reverse the process. A positive power moves the decimal point right, and a negative power moves it left, filling gaps with zeros.

Worked example 3

Write as ordinary numbers: (a) 6.08 × 10⁴ (b) 2.03 × 10⁻³.

(a) Move the point 4 places right: 60,800.

(b) Move the point 3 places left: 0.00203.

Multiplying and Dividing

Deal with the numbers and the powers of 10 separately, using the index laws 10ᵃ × 10ᵇ = 10^(a+b) and 10ᵃ ÷ 10ᵇ = 10^(a−b). Then adjust the answer so that A is between 1 and 10.

Worked example 4

Evaluate, giving answers in the form A × 10ⁿ:

(a) (3 × 10⁴) × (5 × 10⁶) (b) (8.4 × 10⁷) ÷ (2.1 × 10³) (c) (6 × 10⁻³) × (4 × 10⁵) (d) (2.4 × 10⁻²) ÷ (8 × 10⁻⁶)

(a) 3 × 5 = 15 and 10⁴ × 10⁶ = 10¹⁰, giving 15 × 10¹⁰. Since 15 = 1.5 × 10¹, the answer is 1.5 × 10¹¹.

(b) 8.4 ÷ 2.1 = 4 and 10⁷ ÷ 10³ = 10⁴, giving 4 × 10⁴.

(d) 2.4 ÷ 8 = 0.3 and 10⁻² ÷ 10⁻⁶ = 10⁴, giving 0.3 × 10⁴. Since 0.3 = 3 × 10⁻¹, the answer is 3 × 10³.

When adjusting: if A becomes 10 or more, divide A by 10 and add 1 to the power. If A becomes less than 1, multiply A by 10 and subtract 1 from the power.

Adding and Subtracting

You cannot simply add the A values unless the powers of 10 are the same. Either convert to ordinary numbers, or rewrite one number so both have the same power.

Worked example 5

Evaluate (a) 3.2 × 10⁴ + 5.6 × 10³ (b) 4.5 × 10⁻³ − 8 × 10⁻⁴.

(a) Ordinary numbers: 32,000 + 5,600 = 37,600 = 3.76 × 10⁴.

Alternatively, write 5.6 × 10³ as 0.56 × 10⁴. Then (3.2 + 0.56) × 10⁴ = 3.76 × 10⁴.

(b) 0.0045 − 0.0008 = 0.0037 = 3.7 × 10⁻³.

Standard Form and Significant Figures

Such answers are often rounded to a given number of significant figures. Round A, and keep the power of 10 the same.

Worked example 6

Write 0.0045678 as A × 10ⁿ correct to 3 significant figures.

0.0045678 = 4.5678 × 10⁻³, and 4.5678 rounds to 4.57, so the answer is 4.57 × 10⁻³.

WAEC-Style Questions

Worked example 7

Evaluate (0.0024 × 0.0035) ÷ 0.042, leaving your answer in standard form.

Write each number as A × 10ⁿ: (2.4 × 10⁻³ × 3.5 × 10⁻³) ÷ (4.2 × 10⁻²).

Numerator: 2.4 × 3.5 = 8.4, and 10⁻³ × 10⁻³ = 10⁻⁶, giving 8.4 × 10⁻⁶.

Divide: 8.4 ÷ 4.2 = 2, and 10⁻⁶ ÷ 10⁻² = 10⁻⁴. The answer is 2 × 10⁻⁴.

Worked example 8

Find √(1.6 × 10⁵) without a calculator.

To take a square root, make the power of 10 even: 1.6 × 10⁵ = 16 × 10⁴.

√16 = 4 and √(10⁴) = 10², so the answer is 4 × 10² = 400.

Worked example 9: a light-year

Light travels at 3 × 10⁸ m/s, and a year has about 3.15 × 10⁷ seconds. Find the distance light travels in one year.

Distance = speed × time = 3 × 10⁸ × 3.15 × 10⁷ = 9.45 × 10¹⁵ m. This distance, about 9.45 × 10¹⁵ m, is called a light-year.

Worked example 10: ordering numbers

Arrange in ascending order: 4.2 × 10⁻³, 3.9 × 10⁻², 5.1 × 10⁻⁴, 2.8 × 10⁻³.

Compare the powers first (the more negative the power, the smaller the number), then compare A when the powers are equal.

Ascending order: 5.1 × 10⁻⁴, 2.8 × 10⁻³, 4.2 × 10⁻³, 3.9 × 10⁻².

Using a Calculator

Scientific calculators have an EXP or ×10ˣ key. To enter 3.2 × 10⁴, press 3.2, then EXP, then 4. Calculators often display very large or small answers as something like 1.5E11, meaning 1.5 × 10¹¹. When you write the answer, always convert this display into proper standard form notation.

Common Mistakes with Standard Form

Leaving A outside the range, such as writing 24 × 10² instead of 2.4 × 10³.
Getting the sign of the power wrong. Numbers less than 1 always have a negative power.
Adding A values with different powers, such as 3.2 × 10⁴ + 5.6 × 10³ = 8.8 × 10⁴ (wrong).
Miscounting zeros in long numbers. Count carefully, or group digits in threes.

Standard Form: Practice Questions

Write 0.000506 and 82,400,000 in standard form.
Write 3.07 × 10⁻⁴ as an ordinary number.
Evaluate (5 × 10³) × (7 × 10⁻⁵).
Evaluate (9.6 × 10⁸) ÷ (3.2 × 10⁻²).
Evaluate 6.1 × 10⁵ − 4 × 10⁴.
Write 0.00098765 as A × 10ⁿ correct to 2 significant figures.

Answers: 1. 5.06 × 10⁻⁴ and 8.24 × 10⁷. 2. 0.000307. 3. 35 × 10⁻² = 3.5 × 10⁻¹. 4. 3 × 10¹⁰. 5. 610,000 − 40,000 = 570,000 = 5.7 × 10⁵. 6. 9.9 × 10⁻⁴.

Conclusion

Standard form writes any number as A × 10ⁿ with 1 ≤ A < 10. Large numbers have positive powers and small numbers have negative powers. To multiply or divide, work with the numbers and the powers separately, then adjust A back into range. To add or subtract, first make the powers equal or convert to ordinary numbers. With this skill, the huge and tiny numbers of science, finance and engineering become easy to read, compare and calculate with.