# Scientific Notation

Updated: 14 Aug 2023

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While studying Mathematics and Science, sometimes we face very large and very small numbers with many zeros. Working on such numbers takes time, effort and difficult to solve. To overcome such difficulties, mathematicians have developed a technique called scientific notation, using which we can write very large and very small numbers in a more concise and manageable form by expressing them as a product of a coefficient and a power of 10.

## Scientific Notation

Scientific notation is a way of writing numbers that are too big or too small to be easily written in decimal form.

### Scientific Notation Explanation

The positive number "x" is represented in scientific notation as the product of two components; a co-efficient "a" which is a real number greater than 1 and less than 10 and the second is the integral power "n" \ of \ 10 . The general form of a number in scientific notation is x=a×10^n , where a is the coefficient and n is the exponent.

## Standard Form

Standard notation is the everyday way of writing numbers using the base-10 numerical system (0 – 9) . It involves representing numbers as a sequence of digits, where each digit’s position signifies its place value relative to the decimal point.

## Standard form to Scientific Notation

Following are the steps to convert Standard Form to Scientific Notation.

• In a given number, place the decimal after first non-zero digit.
• If the decimal point is moved towards left, then the power of “10” will be positive.
• If the decimal is moved towards right, then the power of “10” will be negative.
• The numbers of digits through which the decimal point has been moved will be the exponent.
##### Write 405,000 in scientific notation.

Solution:
405,000
In Scientific Form:
4.05 \times 10^4

##### Write 0.00092 in scientific notation.

Solution:
0.00092
In Scientific Form:
9.2 \times 10^{-4}

## Scientific Notation to Standard Form

Following are the steps to convert Scientific Notation to Standard Form

• If the exponent of 10 is Positive, move the decimal towards Right.
• If the exponent of 10 is Negative, move the decimal toward Left.
• Move the decimal point to the same number of digits as the exponent of 10.
##### Write 8.3 \times 10^{-5} in standard form.

Solution:
8.3 \times 10^{-5}
In Standard Form:
0.000083

##### Write 4.1 \times 10^6 in standard form.

Solution:
4.1 \times 10^6
In Standard Form:
410000

## Scientific Notation – MCQS

1. __________ is a way of writing numbers that are too big or too small to be easily written in decimal form.
O Standard notation
O Binary notation
O Scientific notation
O All of them

Scientific notation
Explanation:

2. Scientific notation is a way of writing numbers that are too big or too small to be easily written in _________ form.
O Standard
O Binary
O Decimal
O All of them

Decimal
Explanation:

3. General form of scientific notation is
O x=a \times 10^n
O x=a \times 10^a
O x=a \times 10^{10}
O None of them

x=a \times 10^n
Explanation:
Here "a" is real number greater than or equal to 1 but less than 10 and integer power n of 10. i.e.
x=a \times 10^n

4. In standard to scientific notation, the decimal should place after ____________ non – zero digit.
O Zero
O First
O Second
O All of them

First
Explanation:
Rules to convert standard to scientific notation.

5. From standard to scientific notation, if decimal moved towards left, then power of 10 will be ____________
O Positive
O Negative
O Both of them
O None of them

Positive
Explanation:
Rules to convert standard to scientific notation.

6. From standard to scientific notation, if decimal moved towards right, then power of 10 will be ____________
O Positive
O Negative
O Both of them
O None of them

Negative
Explanation:
Rules to convert standard to scientific notation.

7. From scientific to standard notation, it the exponent of 10 is negative, the decimal will move towards ___________
O Left
O Right
O Both of them
O None of them

Left
Explanation:
Rules to convert Scientific to Standard notation.

8. From scientific to standard notation, it the exponent of 10 is positive, the decimal will move towards ___________
O Left
O Right
O Both of them
O None of them

Right
Explanation:
Rules to convert Scientific to Standard notation.

9. The scientific form of 16700000 is
O 1.67 \times 10^7
O 1.67 \times 10^8
O 167.00 \times 10^7
O All of them

1.67 \times 10^7
Explanation:
Place the Decimal after first non-zero digit which is 1.
Count the digits towards left which are 7
Here the decimal moves towards left, So power of 10 is positive. Thus,
1.67 \times 10^7

10. The scientific form of 0.00000039 is
O 3.9 \times 10^7
O 3.9 \times 10^{-7}
O 39 \times 10^8
O 39 \times 10^{-8}

3.9 \times 10^{-7}
Explanation:
Place the Decimal after first non-zero digit which is 3.
Count the digits towards right which are 7
Here the decimal moves towards Right, So power of 10 is Negative. Thus,
3.9 \times 10^{-7}

11. The scientific form of 0.05 \times 10^{-3} is
O 5 \times 10^{-5}
O 5.0 \times 10^{-5}
O 5 \times 10^5
O Both a & b

5.0 \times 10^{-5}
Explanation:
0.05 \times 10^{-3}=5.0 \times 10^{-2} \times 10^{-3}
0.05 \times 10^{-3}=5.0 \times 10^{-2-3}
0.05 \times 10^{-3}=5.0 \times 10^{-5}

12. The standard form of 3.15 \times 10^{-6} is
O 0.0000003
O 0.00000315
O 315
O 3150000

0.00000315
Explanation:
If the Power of 10 is Negative, So Move the decimal towards Left.
Here the power is -6 then move the decimal upto 6 Digits towards left. Thus,
0.00000315

13. The standard form of 3.15 \times 10^{6} is
O 0.0000003
O 0.00000315
O 315
O 3150000

3150000
Explanation:
If the Power of 10 is Positive, So Move the decimal towards Right.
Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,
3150000

14. The standard form of -2.6 \times 10^{6} is
O -2.6
O 0.0000026
O 2600000
O -2600000