Introduction to The Map of Mathematics

OBJECTIVES

The objective of this blog is to familiarize with : -

1.  Introduction to Scientific Notation

2.  Scientific Notation to Standard Form

3.  Decimal Notation to Standard Notation

4.  Scientific Notation Basics

5.  Exponents and Powers of 10 - Basic Rules

6.  Multiplication and Division With Scientific Notation

7.  Scientific Notation - Adding and Subtracting

8.  Square Roots and Radicals

9.  Cube Roots, Perfect Squares, and Cubes

10.  Scientific Notation Negative Exponents

1. Introduction to Scientific Notation:

Scientific notation is a way to write numbers in a compact form that is particularly useful when dealing with very large or very small numbers. It is also known as exponential notation or standard form. It is a method of expressing numbers as the product of a coefficient and a power of ten. The coefficient is usually a decimal number between 1 and 10, and the power of ten indicates the number of places the decimal point must be moved to the left or right to obtain the original number.

 

2 .Scientific Notation to Standard Form:

To convert a number in scientific notation to standard form, we need to multiply the coefficient by the appropriate power of ten. If the power of ten is positive, we move the decimal point to the right. If the power of ten is negative, we move the decimal point to the left. For example, if we have the number 5.6 × 10^3, we can convert it to standard form by multiplying 5.6 by 1000 (which is 10^3), giving us the answer of 5600.

 

3. Decimal Notation to Standard Notation:

To convert a number from decimal notation to scientific notation, we need to move the decimal point to the left or right until the first digit to the left of the decimal point is between 1 and 9. Then we count the number of places we moved the decimal point and use that as the power of ten. For example, the number 0.0000123 can be written in scientific notation as 1.23 × 10^-5, since we moved the decimal point five places to the right to get the number between 1 and 9, and the power of ten is negative because we moved the decimal point to the left.

 

4 .Scientific Notation Basics:

Scientific notation is particularly useful when dealing with very large or very small numbers, as it allows us to write these numbers in a compact form. The exponent in scientific notation tells us the order of magnitude of the number, or how many zeros are after the first significant digit. For example, the number 6.02 × 10^23 represents Avogadro's number, which is the number of atoms or molecules in one mole of a substance. This number is too large to write out in decimal notation, but in scientific notation, we can easily see that it has 23 zeros after the first digit.

 

5. Exponents and Powers of 10 - Basic Rules:

In scientific notation, the exponent represents the number of places the decimal point is moved to the left or right. If the exponent is positive, the decimal point is moved to the right. If the exponent is negative, the decimal point is moved to the left. The basic rules for exponents and powers of ten are:

 

When we multiply numbers in scientific notation, we add the exponents and multiply the coefficients.

When we divide numbers in scientific notation, we subtract the exponents and divide the coefficients.

When we raise a power of ten to a power, we multiply the exponents.

When we take the square root of a number in scientific notation, we divide the exponent by two and take the square root of the coefficient.

6. Multiplication and Division With Scientific Notation:

To multiply or divide numbers in scientific notation, we first multiply or divide the coefficients, and then add or subtract the exponents as appropriate. For example, if we want to multiply 3.5 × 10^4 by 2.1 × 10^3, we can multiply the coefficients (3.5 × 2.1 = 7.35) and add the exponents (4 + 3 = 7), giving us the answer of 7.35

7. Scientific Notation - Adding and Subtracting:

To add or subtract numbers in scientific notation, we first need to make sure the exponents are the same. We can do this by changing the exponent of one or both numbers, by multiplying or dividing the coefficient by a power of ten. Once the exponents are the same, we can add or subtract the coefficients as appropriate. For example, if we want to add 2.3 × 10^4 and 1.8 × 10^4, we can change the exponent of the second number by multiplying both the coefficient and the exponent by 10, giving us 1.8 × 10^5. Now that the exponents are the same, we can add the coefficients (2.3 + 1.8 = 4.1), giving us the answer of 4.1 × 10^4.

8. Square Roots and Radicals:

A square root is the number that, when multiplied by itself, gives a given number. For example, the square root of 25 is 5, because 5 × 5 = 25. In scientific notation, to find the square root of a number, we divide the exponent by two and take the square root of the coefficient. For example, the square root of 4.9 × 10^6 is 7 × 10^2.

A radical is the symbol used to indicate the operation of taking a root. The square root is indicated by the symbol √. For example, √25 = 5.

9. Cube Roots, Perfect Squares, and Cubes:

A cube root is the number that, when multiplied by itself twice, gives a given number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27. In scientific notation, to find the cube root of a number, we divide the exponent by three and take the cube root of the coefficient. For example, the cube root of 1.331 × 10^3 is 1.1 × 10^1.

A perfect square is a number that is the square of an integer. For example, 9 is a perfect square because it is equal to 3 × 3. In scientific notation, a perfect square can be written as the square of a number in scientific notation. For example, 4.9 × 10^6 is a perfect square because it is equal to (7 × 10^2)^2.

A cube is a number that is the cube of an integer. For example, 8 is a cube because it is equal to 2 × 2 × 2. In scientific notation, a cube can be written as the cube of a number in scientific notation. For example, 1.331 × 10^3 is a cube because it is equal to (1.1 × 10^1)^3.

10. Scientific Notation Negative Exponents:

A negative exponent in scientific notation indicates that the number is a fraction or a very small number. For example, 1.23 × 10^-5 represents the number 0.0000123. To convert a number with a negative exponent to standard form, we need to move the decimal point to the left by the absolute value of the exponent. For example, to convert 1.23 × 10^-5 to standard form, we need to move the decimal point five places to the left, giving us the answer of 0.0000123.

When we multiply a number in scientific notation by 10^n, we add n to the exponent. For example, 1.23 × 10^-5 × 10^3 is equal