The Decimal Number System

1
The Decimal Number System

• Decimal Number System
• Decimal Fractions
• Rounding Whole Numbers
• Rounding Non-whole Numbers

2
Decimal Numbers (cont.)

• Signed Numbers
• Addition And Subtraction
• Multiplication And Division
• Mathematical Expressions And Terms

3
Decimal Number System (1)

• Decimal means base ten
• The decimal system uses 10 digits
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Each position in a number has a place value.

4
Decimal Number System (2)

• The MSD is leftmost nonzero digit.
• The MSD (most significant digit) has the greatest
  effect on the value of the number.
• The least significant digit (LSD) is the digit
  with a place value that has the smallest effect
  upon the number's value. The LSD is at the right
  of a number.
•  The LSD can be (and often is) zero.

5
Decimal Fractions (1)

• A fraction has two numbers (1/2). The numerator,
  and the denominator.
• In a fraction, the numerator is divided by the
  denominator.
• A fraction whose denominator is 10 or a power of
  ten is known as a decimal fraction.
• To convert from a decimal fraction to a decimal
  number, determine the value of the denominator
  and place the LSD of the numerator in that
  position.

6
Rounding Whole Numbers (1)

• Rounding numbers to the nearest place value,
  which becomes the new LSD, is a common method
  used to simplify calculations.
• A general rule is if the digit to the right of
  the new LSD is 4 or smaller (4, 3, 2, 1, 0),
  replace the digits to the right of the new LSD
  with zeros.
• if the digit to the right of the new LSD is 5 or
  greater (5, 6, 7, 8, 9), increase the new LSD by
  one and replace the digits to the right of the
  new LSD with zeros.

7
Rounding Whole Numbers (2)

• In order to maintain the desired number of
  significant digits, truncate the number at the
  new LSD and add a times value or words so that
  the number of significant digits is correct.
• If you want 3 significant digits, then 654321
  becomes 654000 which becomes 654 x 1000 or 654
  thousand.
• 3456789 becomes 3460000 which becomes 346 x 10000
  or 346 ten thousands

8
Rounding Non-whole Numbers

• Use the same system as above to round decimal
  numbers. The number of significant digits is
  independent of decimal place. If you want 4
  significant digits, 0.00345678 becomes 0.003457
  and 876.54321 becomes 876.5

9
Signed Numbers

• Numbers can have or - signs. These numbers are
  called signed numbers.
• Some common symbols are lt (less than) and gt
  (greater than). For example, 1 lt 2 (1 is less
  than 2) and 6.7 gt 5.2 (6.7 is greater than 5.2).
• If the value of the number is used without regard
  to the sign, then that value is called the
  absolute value. The absolute value of -17 and 17
  is 17. The magnitude of a number ignores the
  sign.

10
Addition And Subtraction (1)

• The signs and - indicate the sign of a number.
  They also indicate addition and subtraction. The
  magnitude of the number is the number without a
  sign.
• Adding two or more positive numbers involves just
  adding the magnitudes and using a plus () sign
  on the result.

11
Addition And Subtraction (2)

• Adding two or more negative numbers involves just
  adding the magnitudes and using a negative sign
  (-) on the result.
• Adding numbers with opposite signs is a different
  process. Add together all the positive numbers
  and then add together all the negative numbers.
  Subtract the absolute value of the lesser number
  from the absolute value of the greater number and
  place the sign of the largest number before the
  answer.

12
Addition And Subtraction (3)

• The process of subtraction involves the first
  number called the minuend, minus the second
  number, called the subtrahend. The answer is
  called the difference. The rule in subtraction is
  to change the sign of the subtrahend and add as
  before.

13
Multiplication And Division (1)

• Symbols are used to denote multiplication and
  division. The symbol for multiplication is x or
  or no operator given before a left paren (or a
  letter).
• Division symbols are ? or / or a fraction line.
  Division always means to take reciprocal and then
  multiply.

14
Multiplication And Division (2)

• Multiplying or dividing signed numbers requires a
  method of determining the sign of the result.
• Positive signed numbers multiplied or divided
  will always result in a positive answer.
• Multiplying or dividing an even number of
  negative numbers will result in a positive
  answer. Multiplying or dividing an odd number of
  negative numbers will result in a negative answer.

15
Mathematical Expressions And Terms (1)

• A mathematical term is a number preceded by a
  or - sign.
• An expression is a group of two or more terms.
• Mathematical expressions are often grouped
  together in parenthesis ( ), brackets J, or
  braces . All of these are types of parentheses.

16
Mathematical Expressions And Terms (2)

• Use the mnemonic Please Excuse My Dear Aunt Sally
  to remember the order of operations.
• A mathematical expression using these groupings
  must be solved by first performing the
  calculations within each set of parentheses,
  brackets, or braces.
• Next do any exponents (which includes powers and
  trig functions).
• Change any number after division sign to the
  reciprocal and change ? to x since division is
  just a special type of multiplication.

17
Mathematical Expressions And Terms (3)

• Next do all multiplications (including those from
  reciprocals of division).
• Lastly, do all additions and subtractions.
• The End