The Decimal Number System Decimal Fractions Rounding Whole Numbers Rounding Non‑whole Numbers

Decimal Numbers (cont.) Signed Numbers Addition And Subtraction Multiplication And Division Mathematical Expressions And Terms

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.

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.

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.

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.

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

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

Signed Numbers Numbers can have + or ‑ signs. These numbers are called signed numbers. Some common symbols are < (less than) and > (greater than). For example, 1 < 2 (1 is less than 2) and 6.7 > 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.

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.

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.

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.

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.

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.

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.

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.

Mathematical Expressions And Terms (3) Next do all multiplications (including those from reciprocals of division). Lastly, do all additions and subtractions. The End