Civil Service Exam (Subprofessional) Numerical Ability — Decimals & Scientific NotationStudy Notes

A decimal is a way of representing numbers that are not whole using a decimal point. The decimal system is based on powers of 10, where each position to the right of the decimal point represents a fraction with a denominator that is a power of 10. For example, in the number 12.345, the digit 3 is in the tenths place (3/10), 4 is in the hundredths place (4/100), and 5 is in the thousandths place (5/1000). Understanding place values is crucial for reading, writing, and performing operations with decimals. When reading decimals aloud, we say the whole number part, then 'and' for the decimal point, followed by the decimal part and its place value. For instance, 176.304 is read as 'one hundred seventy-six and three hundred four thousandths'.

Adding and subtracting decimals requires careful alignment of decimal points to ensure proper place value calculations. The key is to write all numbers vertically with decimal points aligned, add zeros as placeholders when necessary, and perform the operation as you would with whole numbers. After computation, place the decimal point in the answer directly below the aligned decimal points in the problem. This method ensures that digits of the same place value are correctly combined.

Multiplying decimals follows a systematic approach where you first ignore the decimal points and multiply as if dealing with whole numbers. After obtaining the product, count the total number of decimal places in both factors and place the decimal point in the product so that it has the same total number of decimal places. For example, when multiplying 0.25 × 0.5, multiply 25 × 5 = 125, then count decimal places: 0.25 has 2 decimal places, 0.5 has 1 decimal place, so the product must have 3 decimal places, giving us 0.125.

Dividing decimals requires making the divisor a whole number by moving its decimal point to the right. You must move the dividend's decimal point the same number of places to maintain the quotient's value. Place the decimal point in the quotient directly above the decimal point in the adjusted dividend, then divide as you would with whole numbers. If there's a remainder, you can add zeros to continue division until you reach the desired precision or recognize a repeating pattern.

Converting fractions to decimals involves dividing the numerator by the denominator. Some fractions convert to terminating decimals (like 1/4 = 0.25), while others result in repeating decimals (like 1/3 = 0.333...). Understanding this conversion is crucial for working with mixed problems involving both fractions and decimals. To convert, simply perform the division, adding decimal places as needed. For repeating decimals, you can round to a specified number of decimal places or use the repeating decimal notation.

Scientific notation is a method of expressing very large or very small numbers in a compact form using powers of 10. It's written as a × 10^n, where 'a' is a number between 1 and 10 (including 1 but not 10), and 'n' is an integer that indicates how many places the decimal point has been moved. For large numbers, the exponent is positive; for small numbers, the exponent is negative. This notation is essential in science, engineering, and technology for handling measurements that range from astronomical distances to atomic dimensions.

To convert a number to scientific notation, move the decimal point until you have a number between 1 and 10. Count how many places you moved the decimal point - this becomes your exponent. If you moved the decimal point to the left (for large numbers), the exponent is positive. If you moved it to the right (for small numbers), the exponent is negative. The original number's significant digits become the coefficient in your scientific notation.

When performing operations with numbers in scientific notation, you handle the coefficients and exponents separately. For multiplication, multiply the coefficients and add the exponents. For division, divide the coefficients and subtract the exponents. For addition and subtraction, the numbers must have the same exponent, so you may need to adjust one number before performing the operation. Always express your final answer in proper scientific notation form.

Decimals and scientific notation have numerous practical applications in everyday life and professional fields. In finance, decimals are used for currency calculations, interest rates, and percentage calculations. Scientific notation is essential in physics for expressing constants like Planck's constant (6.626 × 10⁻³⁴), in astronomy for distances between stars, in chemistry for molecular masses, and in technology for data storage capacities. Understanding these concepts helps in solving problems related to measurements, conversions, and calculations in various scientific and practical contexts.