1 .1 – THE CONCEPT OF OPPOSITES

Any movement from an initial point on the number line going to the right is represented by a positive sign (+) ,while a movement to the left is represented by a negative sign (-).These numbers are sometimes called directed numbers or signed numbers. e.g.

1.2 - FUNDAMENTAL OPERATIONS ON INTEGERS

1.2.1 – ADDITION OF INTEGERS

To add integers with the same sign ,add without regard to the signs.Then affix the common sign of the integers.To add two integers with different signs ,consider the distance of each integer from zero (that is, consider the absolute value of each addend).Subtract the shorter distance from the longer distance. In the answer ,use the sign of the number farther from zero. e.g. 82 + (-62) + 29 + (-25) = (82 + 29) + (-62 + -25) = 111 + (-87) = 24 1.2.2 – SUBTRACTION OF INTEGERS

To find the difference between two signed numbers ,add the negative (or the opposite) of the subtrahend to the minuend. e.g. a. -62 – (-135) = -62 + 135 = 73 b. 29 - 86 = 29 + (-86) = -57 1.2.3 – MULTIPLICATION OF THE INTEGERS

The product of two integers with the same sign is positive.The product of two integers with different signs is negative. e.g. a. -20 x 15 = -300 b. -15 x (-35) = 525 c. -45 x (-45) = -2025 1.2.4 – DIVISION OF INTEGERS

The quotient of two integers with the same sign is positive. The quotient of two integers with different signs is negative. e.g. a. 165 ÷ 15 = 11 b. -180 ÷ (-12) = 15 c. -990 ÷ 22 = -45 1.3 – THE ABSOLUTE VALUE OF A NUMBER

The absolute value of a number is the distance on the number line between the number and zero without any regard to its direction.Thus ,the absolute value of any number is a nonnegative number. e.g. ǀ-10ǀ + ǀ15ǀ - ǀ6ǀ = 10 + 15 -6 = 25 – 6 = 19 1.4 – OPERATIONS ON FRACTIONS

1.4.1 – ADDITION AND SUBTRACTION OF FRACTIONS

If a ,b ,and c are integers ,where b ҂ 0 ,then ab + cb = a+cb . e.g. 2 12 + 1 12 = 52 + 32 = 5+32 = 82 = 4

If a ,b ,and c are integers and b ҂ 0 ,then ab - cb = a-cb . e.g. 10 34 – 2 38 = 434 - 198 = 868 - 198 = 678 or 8 38 1.4.2 – MULTIPLICATION AND DIVISION OF FRACTIONS

If a ,b ,c ,and d are integers ,where b ҂ 0 and d ҂ 0 ,then ab x cd = a x cb x d . Every rational number except 0 has a unique reciprocal.

e.g. 35 x 23 = 3 x 25 x 3 = 615 or 25

If a ,b ,c ,and d are integers ,where b ҂ 0 ,c ҂ 0 and d ҂ 0 ,then ab ÷ cd = a x db x c . e.g. 6 23 ÷ -112 = 203 ÷ -32 = 203 x -23 = 20 x (-2)3 x 3 = -409 or -449 1.5 – OPERATIONS ON DECIMALS

1.5.1 – ADDITION AND SUBTRACTION OF DECIMALS

To add or subtract decimals ,arrange the digits in column according to their corresponding place values and add or subtract as with whole numbers. e.g. a. 0.005 + 9.684 = 9.689 b. 16.4831 – 3.121 = 13.3621 1.5.2 – MULTIPLICATION OF DECIMALS

To multiply decimals ,multiply as in whole numbers.Then ,count the total number of decimal places to the right-hand side of the decimal point in the factors.The product should have this number of decimal places. e.g. 56.8 x 0.63 or 60 x 0.6 = 36.0 or 36

2.5.3 – DIVISION OF DECIMALS

To divide decimals ,multiply both the divisor and the dividend by the same multiple of 10 to make a whole number divisor.Then divide as in whole numbers.Put the decimal point in the quotient directly above the decimal point in the divide. e.g 3.66 ÷ 23.5 = 0.56

2.6 – PROPERTIES OF REAL NUMBERS

2.6.1 – CLOSURE PROPERTY

If a and b are real numbers ,then a + b is also a real number. If a and b are real numbers ,then ab is also a real number. e.g. The sum of the real numbers 8 and 15 is 13 ,which is also a real number. 2.6.2 – COMMUTATIVE PROPERTY

For any real numbers a and b ,we have a + b = b + a.

For any real numbers a and b ,we have ab = ba....