Scientific notation

The idea

In scientific notation, or standard form, a number is written as x × 10ʸ, where the coefficient x satisfies x ≥ 1 and x < 10 and the exponent y is any integer.

Too many zeros to count

The number of stars in our galaxy is 1,00,00,00,00,000.

It is easy to miscount the zeros, like getting ₹5,000 when you were supposed to get ₹50,000.

How many zeros come after the 1?

• 9
• 11
• 13

Slide the decimal point

Slide the point. The value does not change: each step left makes the power of 10 bigger by 1. Stop when x is between 1 and 10.

Scientific form: x × 10ʸ

In scientific notation, or standard form, a number is written as x × 10ʸ, where the coefficient x ≥ 1 and x < 10, and the exponent y is any integer.

5900 = 590 × 10^1 = 59 × 10^2 = 5.9 × 10^3 = 0.59 × 10^4

Only 5.9 × 10^3 has x between 1 and 10.

20800 = 2.08 × 10^4 80,00,000 = 8 × 10^6

Notes

x × 10ʸ with 1 ≤ x < 10 and y an integer is the scientific form (standard form) of a number.

Check yourself

Which is the scientific form of 20800?

Write 308100000 as 3.081 × 10 raised to which power? (Type the exponent.)

Answer: 8

3,08,100,000 has 9 digits. After the first digit, 8 more digits follow, so the exponent is 8.

Write 4.5 × 10^3 as an ordinary number.

Answer: 4500

4.5 × 1000 = 4500.

Which one is not in scientific form?

• 20.8 × 10^3. The number part 20.8 is 10 or more. It must be less than 10.
• 2.08 × 10^4 — correct. Yes! x = 2.08 is between 1 and 10, and 2.08 × 10000 = 20800.
• 0.208 × 10^5. The number part 0.208 is less than 1. It must be at least 1.
• 3.5 × 10^4. x = 3.5 is between 1 and 10, so this is in scientific form.
• 12 × 10^5 — correct. Yes! x = 12 is 10 or more. In scientific form it is 1.2 × 10^6.
• 9.99 × 10⁻². x = 9.99 is between 1 and 10 and the exponent can be any integer, even a negative one.