Mathematics Fundamentals

Study Notes

Adding and Subtracting Fractions

Adding and subtracting fractions involve combining or separating equal parts of a whole. To perform these operations, ensure the denominators are the same, either by multiplying both numerator and denominator by the same number (i.e., the least common multiple of the denominators) or by dividing all components of both fractions by the same factor (i.e., the greatest common divisor of the denominators). Here are some examples:

1 + 1/2 = 2/2 = 1
1 - 1/2 = 1/2
3/4 + 1/2 = 7/4
(1/2) - (3/4) = (-3)/4

Number Notation

Number notation refers to the symbols used to represent numbers. Different cultures worldwide developed their own unique systems of representing numbers, including the Babylonian sexagesimal system, the Chinese decimal system, the Greek and Roman duodecimal system, and the Hindu decimal system. Today, we primarily use the decimal system, which includes ten digits from 0 through 9. Arabic numerals are used universally for their relative simplicity and lack of ambiguity compared to other systems.

Factors and Multiples

Factors are the numbers that can be multiplied together to produce a given number. The prime factorization of a number breaks it down into its smallest distinct prime factors. For example, the prime factorization of 12 is 2 × 2 × 2 × 3, since 2 × 2 × 2 × 3 = 12. A composite number is a number that has factors other than 1 and itself. For example, 12 has the factors 1, 2, 3, 4, 6, and 12.

Multiples are the result of multiplying a single number by some integer, resulting in a new set of values. For example, the multiples of 8 are 8, 16, 24, 32, etc. The multiples of 9 are slightly different because there is a pattern: 9, 18, 27, 36, ..., with each successive multiple increasing by 9.

Division Algorithm

The division algorithm states that every nonzero integer divides another integer exactly once according to the rule: Divide the larger integer by the smaller integer to get a quotient and remainder. If the larger integer is also positive, the quotient will be the largest possible integer less than or equal to the actual quotient, and the remainder will be the actual remainder.

For example, if we divide 12 by 4, we get the quotient of 3 and the remainder of 0, indicating that 12 can be expressed as 4 × 3 + 0. Similarly, if we divide 15 by 3, we get the quotient of 5 and the remainder of 0, allowing us to represent 15 as 3 × 5 + 0.

Mixed Numbers

A mixed number represents a fraction whose numerator is part of a whole number and the denominator is a whole number. To convert a mixed number to an improper fraction, simply multiply the denominator by the whole number and add the numerator to the product. Then, adjust the denominator to remove any unnecessary factors and simplify if possible.

For example, to convert 3 1/2 to an improper fraction, start by multiplying the denominator (2) by the whole number (3): 3(2) = 6. Add the numerator (1) to the product: 6 + 1 = 7. Simplifying gives us the improper fraction of 7/2.

Description

Explore essential mathematical concepts including adding and subtracting fractions, number notation systems, factors and multiples, the division algorithm, and converting mixed numbers to improper fractions. Enhance your understanding of fundamental math principles with examples and explanations.