Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Number Systems → Exponents and Radicals ↓


Scientific Notation


Scientific notation is a method of writing very large or very small numbers in a concise form. This form makes it easier to perform calculations and conceptualize extreme values. Scientific notation expresses numbers as multiples of two factors: the number between 1 and 10 and the power of ten. It is widely used in mathematics, physics, engineering, and other sciences.

Understanding scientific notation

Scientific notation is expressed as follows:

a × 10^n

Where:

  • a is a number that is greater than or equal to 1 and less than 10.
  • n is an integer, which can be positive, negative, or zero.

For example, the number 3000 can be written in scientific notation like this:

3 × 10^3

Why use scientific notation?

Scientific notation is especially useful when working with very large numbers such as the speed of light (about 299,792,458 meters per second) or very small numbers such as the mass of an electron (about 0.00000000000000000000000000000091093822 kilograms).

Writing these numbers in decimal form is cumbersome and even more challenging to calculate. Scientific notation reduces this complexity and provides a convenient way to perform multiplication and division, especially using a calculator or computer.

How to convert to scientific notation

Follow these simple steps to convert a number to scientific notation:

  1. Move the decimal point in the number so that only the first non-zero digit remains to the left of the decimal point.
    For example, to convert 12300 to scientific notation: 
    1.23
  2. Count the number of places, p, that you moved the decimal point.
    In this example, you moved the decimal point 4 places to the left.
  3. Express the original number as the product of a and the power p of 10. 
    1.23 × 10^4

Examples of scientific notation

Let us work through some examples to understand this concept better:

Example 1: Large numbers

Convert 5,800,000 to scientific notation:

  1. Move the decimal point so that there is only one digit to the left: 
    5.8
  2. Count the moves: it's 6 moves to the left.
  3. Write it like this 
    5.8 × 10^6

Example 2: Small number

Convert 0.00045 to scientific notation:

  1. Move the decimal point so that there is only one digit to the left: 
    4.5
  2. Count the moves: It's 4 moves to the right.
  3. Write it like this 
    4.5 × 10^-4

Operations with scientific notation

When performing operations such as multiplication and division with numbers in scientific notation, you can take advantage of the properties of exponents:

Multiplication

When multiplying numbers in scientific notation:

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

Example:

Multiply (3 × 10^4) and (2 × 10^2) :

(3 × 10^4) × (2 × 10^2) = 6 × 10^(4+2) = 6 × 10^6

Division

When dividing numbers in scientific notation:

(a × 10^n) ÷ (b × 10^m) = (a ÷ b) × 10^(n-m)

Example:

Divide (6 × 10^6) by (2 × 10^2) :

(6 × 10^6) ÷ (2 × 10^2) = 3 × 10^(6-2) = 3 × 10^4

Additional examples

Here are some more conversions and calculations with scientific notation to help you practice:

Conversion to scientific notation

  • 0.00567 in scientific notation 
    5.67 × 10^-3
  • 987,000 in scientific notation is 
    9.87 × 10^5
  • 0.0000002085 in scientific notation is 
    2.085 × 10^-7

Performing the calculation

Calculate the result using scientific notation:

  1. (2.5 × 10^5) + (3.5 × 10^5) = ... 
    Align exponents: (2.5 + 3.5) × 10^5 = 6.0 × 10^5
  2. (6.3 × 10^7) × (4.2 × 10^-3) = ... 
    (6.3 × 4.2) × 10^(7 – 3) = 26.46 × 10^4 = 2.646 × 10^5

Practicing with exponents and radicals

Scientific notation often involves operations with exponents. Let's explore these with practical examples:

Exponential

It is extremely important to understand and use powers of ten in scientific notation:

  1. 10^3 means 10 × 10 × 10 = 1,000
  2. 10^-4 means 1/(10 × 10 × 10 × 10) = 0.0001

Radicals

Scientific notation can simplify roots, especially square roots and higher-order roots:

Example: Simplifying the square root of 2,500,000

2,500,000 = 2.5 × 10^6
√(2.5 × 10^6) = √2.5 × √10^6
= 1.58 × 10^3

Visual example: powers of ten

1 × 10^0 (1) 1 × 10^1 (10) 1 × 10^2 (100) 1 × 10^3 (1,000) 1 × 10^4 (10,000) 1 × 10^5 (100,000) 1 × 10^-1 (0.1) 1 × 10^-2 (0.01) 1 × 10^-3 (0.001)

Key takeaways

  • Scientific notation makes it simple to work with large and small numbers.
  • It is expressed as a × 10^n.
  • Exponents in scientific notation help multiply and divide numbers effectively.
  • For practice, make sure you're comfortable converting and working with scientific notation.

Conclusion

Scientific notation is a powerful tool for efficiently handling exceptionally large or small numbers in math and science. Mastering scientific notation ensures better understanding in dealing with data that requires precision or concerns high magnitude. Extensive practice on performing conversions and calculations with scientific notation will lead to greater confidence and competence in applications involving complex calculations.


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