Number Systems and Arithmetic Operations.pptx

Math

1. Number System & Basic Arithmetic Operations Sana Khurshid Nursing Lecturer

2. The Decimal Number System The decimal number system, also known as the base-10 system, is the most common numeral system used. It is based on ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These symbols are called digits.

3. For example, in the decimal number 456: The digit 6 is in the "ones" place, representing 6 units (10^0). The digit 5 is in the "tens" place, representing 5 tens (10^1). The digit 4 is in the "hundreds" place, representing 4 hundreds (10^2). By combining these digits according to their place values, we get the value of the entire number: 400 + 50 + 6 = 456.

4. Decimal to Binary To convert a decimal number to binary, repeatedly divide the decimal number by 2 and note down the remainders. 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1

5. Decimal to Octal To convert a decimal number to octal, repeatedly divide the decimal number by 8 and note down the remainders. The octal representation will be the sequence of remainders read from bottom to top. Example: Convert decimal 42 to octal: 42 ÷ 8 = 5 remainder 2 5 ÷ 8 = 0 remainder 5

6. Whole Numbers Whole numbers are a set of numbers that includes all the non-negative integers, starting from zero and extending infinitely in the positive direction. Whole numbers are represented by the set {0, 1, 2, 3, ...}. They do not include fractions or decimals.

7. Integers Integers are a set of numbers that includes all the whole numbers (positive, negative, and zero) as well as their negative counterparts. Integers are represented by the set {..., -3, -2, -1, 0, 1, 2, 3, ...}. They include both positive and negative whole numbers, as well as zero.

8. Rational Numbers Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Rational numbers include integers, as any integer can be expressed as a fraction with a denominator of 1. Rational numbers may be terminating (finite) decimals or repeating (infinite) decimals. They are represented as fractions or as decimals that either terminate or repeat. Examples of rational numbers include 1/2, -3, 0, 0.75, -2.5, 1.333..., etc.

9. Basic Arithmetic Operations Addition Subtraction Multiplication Division

10. Addition Whole Numbers: Example: 3 + 5 = 8 Integers: Example: (-3) + 5 = 2 Rational Numbers: Example: 1/2 + 3/4 = 5/4 (or 1.25 as a decimal)

11. Subtraction Whole Numbers: Example: 8 - 3 = 5 Integers: Example: 5 - (-3) = 8 Rational Numbers: Example: 3/4 - 1/2 = 1/4 (or 0.25 as a decimal)

12. Multiplication Whole Numbers: Example: 4 * 7 = 28 Integers: Example: (-2) * 3 = -6 Rational Numbers: Example: 1/2 * 3/4 = 3/8 (or 0.375 as a decimal)

13. Division Whole Numbers: Example: 15 ÷ 3 = 5 Integers: Example: 10 ÷ (-2) = -5 Rational Numbers: Example: 3/4 ÷ 1/2 = (3/4) * (2/1) = 3/2 (or 1.5 as a decimal)

14. Fractions A fraction represents a part of a whole or a ratio of two quantities. It consists of two numbers separated by a horizontal or diagonal line, called the fraction bar or division bar. The number above the line is called the numerator, and the number below the line is called the denominator. Fractions can be proper (where the numerator is less than the denominator), improper (where the numerator is greater than or equal to the denominator), or mixed numbers (a whole number combined with a proper fraction). Example: 3/4​ represents three parts out of four equal parts.

15. Decimals Decimals are a way to represent parts of a whole number or fractions using a decimal point. They consist of digits (0-9) and a decimal point, which separates the whole number part from the fractional part. The digits to the left of the decimal point represent the whole number part, and the digits to the right of the decimal point represent the fractional part. Decimals can be finite (terminating) or infinite (repeating). Example: 0.75 represents seventy-five hundredths, which is equivalent to 3/4​.

16. Convert between fractions & decimals Converting Fractions to Decimals: To convert a fraction to a decimal, divide the numerator by the denominator. Example: Convert 3/4​ to a decimal: 3/4=3÷4=0.7543​=3÷4=0.75

17. Cont. Converting Decimals to Fractions: To convert a decimal to a fraction, write the decimal as a fraction with the same value and simplify if necessary. Example: Convert 0.5 to a fraction: 0.5 can be written as 5/10​. Since both 5 and 10 can be divided by 5, the simplified fraction is 1/2

18. Arithmetic operations with fractions & decimals Addition + Subtraction Multiplication Division  

19. Percentages Percentages are a way of expressing a proportion or a ratio as a fraction of 100. The term "percent" means "per hundred," and percentages are commonly denoted by the symbol "%".

20. Relation to Fractions Example: 3/5​ as a percentage is 3/5×100. Relation to Decimals 0.75 as a percentage is 0.75×100.

21. Scenarios In a class of 30 students, 20% are studying mathematics. How many students are studying mathematics? You have a meal at a restaurant that costs $60, and you want to leave a 15% tip. How much should you tip?

22. Exponents & Powers Exponents: An exponent is a small number written above and to the right of a base number. It indicates how many times the base number is multiplied by itself. In the expression 2 3 , the base is 2, and the exponent is 3. It means that 2 is multiplied by itself 3 times, resulting in 2×2×2. Powers: A power is the result of raising a base number to an exponent. It represents the value obtained by repeated multiplication of the base. In the expression 2 3 , the power is 8, because 2 3 =2×2×2=8.

23. Examples 2 3 ×2 4 5 6 ÷5 3 (3 2 ) 4 2 −3 =1/2 3 ​= 1/ 8

24. Apply the Order of Operations 5+3×2 (8−2) 2 ÷2 (8−2) 2 4×(6−3) 2 +5 10−2×3+4 8÷2(2+2)

25. Prime Numbers A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.

26. Composite Numbers A composite number is a natural number greater than 1 that has more than two distinct positive divisors. The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, and so on.

27. Prime Factorization of Composite Numbers Suppose we want to find the prime factors of the composite number 48.

28. Any question?