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Scientific Notation Calculator

Convert numbers to and from scientific notation (standard form), and perform addition, subtraction, multiplication and division in standard form. Full step-by-step working shown for every operation.

Enter any positive or negative number including decimals.

Enter a (coefficient) and n (exponent) for a × 10n

Enter two numbers in scientific notation (a × 10n) and choose an operation.

First number (a × 10n)

Second number (b × 10m)

What Is Scientific Notation? (Standard Form in UK GCSE)

Scientific notation and standard form are the same thing. In the United Kingdom, the GCSE mathematics syllabi — including AQA, Edexcel and OCR — use the term standard form, while the rest of the world (and scientists globally) say scientific notation. Both mean expressing a number as:

a × 10n

where 1 ≤ a < 10 and n is any integer

The rule that the coefficient a must be between 1 (inclusive) and 10 (exclusive) is essential. Without it, you can write the same number in infinitely many ways. With it, every non-zero number has exactly one standard form representation.

Why Do We Need Scientific Notation?

Science, engineering and mathematics constantly encounter numbers that are either astronomically large or vanishingly small. Writing them out in full is error-prone and cumbersome. Consider:

Ordinary NumberStandard FormContext
602,200,000,000,000,000,000,0006.022 × 1023Avogadro's number (mol−1)
299,792,4583 × 108Speed of light (m/s, approx.)
149,600,000,0001.496 × 1011Earth–Sun distance (m)
0.0000000000011 × 10−12One picometre (atomic scale)
0.000000055 × 10−8Typical atomic radius (m)
0.0000000016021.602 × 10−19Charge of one electron (C)

How to Convert a Number to Standard Form

Follow these steps every time:

  1. Identify the significant figures. Ignore leading zeros and the decimal point position for now.
  2. Place the decimal point after the first non-zero digit. This gives you the coefficient a, where 1 ≤ a < 10.
  3. Count how many places you moved the decimal point. Moving left gives a positive exponent; moving right gives a negative exponent.
  4. Write in the form a × 10n.

Worked Examples

Example 1 — Large number: 45,000

Move the decimal 4 places left: 4.5 × 104

Example 2 — Small number: 0.00073

Move the decimal 4 places right: 7.3 × 10−4

Example 3 — Already between 1 and 10: 6.02

No movement needed: 6.02 × 100

Positive vs Negative Exponents in Standard Form

The sign of the exponent n tells you whether the original number is large or small:

  • Positive exponent (n > 0): The number is greater than 1. The larger n, the larger the number. 106 = 1,000,000.
  • Zero exponent (n = 0): The number is between 1 and 10 (i.e., 100 = 1).
  • Negative exponent (n < 0): The number is between 0 and 1. 10−3 = 0.001.

Students often confuse a negative exponent with a negative number. They are different: 3 × 10−4 = 0.0003 which is positive but small. The number −3 × 104 = −30,000 is negative and large in magnitude.

Multiplying Numbers in Standard Form

Multiplication is the easiest operation in standard form. Use the laws of indices:

(a × 10n) × (b × 10m) = (a × b) × 10n+m

Steps: (1) Multiply the coefficients. (2) Add the exponents. (3) Normalise if the new coefficient is not between 1 and 10.

Example: (3.2 × 104) × (5.0 × 103)

Step 1: 3.2 × 5.0 = 16

Step 2: 104 × 103 = 107

Step 3: 16 × 107 → normalise → 1.6 × 108

Dividing Numbers in Standard Form

(a × 10n) ÷ (b × 10m) = (a ÷ b) × 10n−m

Steps: (1) Divide the coefficients. (2) Subtract the exponents. (3) Normalise.

Example: (8.4 × 106) ÷ (2.0 × 102)

Step 1: 8.4 ÷ 2.0 = 4.2

Step 2: 106 ÷ 102 = 104

Result: 4.2 × 104 (already normalised)

Adding and Subtracting in Standard Form

Addition and subtraction require the powers of 10 to be equal before you can add or subtract the coefficients. This is analogous to adding fractions with the same denominator.

  1. Identify the larger exponent.
  2. Rewrite the number with the smaller exponent so it has the same power of 10 (adjust the coefficient accordingly).
  3. Add or subtract the coefficients.
  4. Normalise if necessary.

Example: 3.2 × 104 + 6.0 × 103

Step 1: Larger exponent is 4. Rewrite 6.0 × 103 as 0.6 × 104.

Step 2: 3.2 × 104 + 0.6 × 104 = 3.8 × 104

Result: 3.8 × 104

Standard Form in GCSE Maths (AQA, Edexcel, OCR)

Standard form appears in every UK GCSE Mathematics specification. It typically accounts for 3–6 marks per exam paper. Key skills assessed include:

  • Converting between ordinary numbers and standard form
  • Ordering numbers given in standard form
  • Multiplying and dividing in standard form (without a calculator)
  • Adding and subtracting in standard form (both with and without a calculator)
  • Interpreting standard form in context (e.g., comparing the size of atoms, planets)

At A-Level, standard form underpins topics including logarithms, exponential functions, and applied science modules. Engineering courses use engineering notation — a variant where the exponent is always a multiple of 3 (e.g., 47,000 = 47 × 103 = 47k), aligning with SI prefixes (kilo, mega, giga, milli, micro, nano).

Scientific Notation on a Calculator

Most scientific calculators display scientific notation using EXP or EE notation. For example, 3.2 × 104 appears as 3.2E4 or 3.2 × 104. Key tips:

  • Press EXP (or EE) to enter the exponent — do not type × 10 separately.
  • In Excel, enter 3.2E4 or use the formula =3.2*10^4. Excel displays it as 3.20E+04 when cells are formatted as Scientific.
  • In Python, 3.2e4 gives 32000.0; 3.2e-4 gives 0.00032.

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Frequently Asked Questions

Scientific notation (called standard form in UK GCSE maths) is a way of writing very large or very small numbers as a × 10n, where 1 ≤ a < 10 and n is an integer. For example, 3,000,000 is written as 3 × 106. The two terms mean exactly the same thing — UK schools use "standard form" while scientists and international curricula say "scientific notation".
Move the decimal point until you have a number between 1 and 10. Count how many places you moved it. If you moved left (large number), the exponent n is positive. If you moved right (small number), n is negative. Example: 45,000 → 4.5 × 104; 0.00073 → 7.3 × 10−4.
Multiply the coefficients and add the exponents: (a × 10n) × (b × 10m) = (a × b) × 10n+m. Then normalise if the result is not in correct standard form. For example: (3 × 102) × (4 × 103) = 12 × 105 = 1.2 × 106.
A positive exponent means a large number (greater than 1). A negative exponent means a small number (between 0 and 1). For example, 106 = 1,000,000 while 10−6 = 0.000001. The sign of the exponent tells you which direction to move the decimal point when converting back to an ordinary number.
You must first make the exponents equal. Adjust one number so both share the same power of 10, then add or subtract the coefficients. Example: 3.2 × 104 + 6 × 103 = 3.2 × 104 + 0.6 × 104 = 3.8 × 104. Always check the result is in correct standard form.
Scientific notation is used wherever numbers are very large or very small: astronomy (Earth–Sun distance = 1.496 × 1011 m), chemistry (Avogadro's number = 6.022 × 1023 mol−1), physics (speed of light ≈ 3 × 108 m/s), biology (size of a bacterium ≈ 1 × 10−6 m), and computing (file sizes, processor clock speeds).
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Mustafa Bilgic

UK Maths Specialist • About UK Calculator • Updated February 2026