http://www.mathsisfun.com/fraction s.html Year 9 Mathematics http://www.mathsisfun.com/fraction s.html Indices

Learning Intentions Understand what is meant by an index Understand the laws for indices Understand the format of a standard form number Be able to change numbers to and from standard form Understand how to add, subtract, multiply and divide numbers in standard form

An Index In mathematics an index is the power that a number is raised to. For example, when we write 23, the number 3 is known as the index. The plural of the word index is indices

Multiplying numbers with Indices When we write 23, we mean 2 x 2 x 2 If we multiply 23 by 24, ie 23 x 24, we mean 2 x 2 x 2 x 2 x 2 x 2 x 2 We could write this as 27 When multiplying numbers with indices we add the indices

Dividing numbers with Indices When we write 25, we mean 2 x 2 x 2 x 2 x 2 If we divide 25 by 22, ie 25 ÷ 22, we mean 2 x 2 x 2 x 2 x 2 ÷ 2 x 2 We could write this as 23 When dividing numbers with indices we subtract the indices

Special Index Sometimes when dividing numbers in index form we end up with a zero index. For example: 23 ÷ 23 = 20 In this case we are dividing a number by itself. In this case the result is always 1 A number with a zero index always equals 1.

Negative Indices When we divide numbers in index form we sometimes end up with a negative index. For example, 23 ÷ 25 = 2-2 Remember: 23 ÷ 25 = = = So 2-2 means , we call this the reciprocal A negative index means we need to find the reciprocal.

Powers Sometimes a number in index form is given an index. For example (23)2 Remember 23 = 2 x 2 x 2 and squaring a number means multiplying it by itself (23)2 = 2 x 2 x 2 x 2 x 2 x 2 = 26 When raising a number to a power, we multiply the indices.

Laws of Indices When multiplying numbers add the indices When dividing numbers subtract the indices A number with a zero index equals 1 Negative indices mean find the reciprocal When numbers in index form are raised to a power, multiply the indices

Standard Form Standard form is used to represent very large and very small numbers. In standard form numbers are always written as: a x 10n The number a is a number between 1 and 10 The number n must always be a whole number and tells us how many places we have moved the decimal point. For example the distance to the moon is 9.3 x 107 miles (93 000 000 miles)

Numbers in Standard Form Let’s look again at the distance to the moon 93 000 000 miles To change this number into standard form we first put in a decimal point so that the number is between 1 and 10. 93 000 000 becomes 9.3 000 000 or 9.3 This gives the value for a

Numbers in Standard Form Next we count the number of places that we moved the decimal point. In this case 7. This gives the value for n We can now write the number in standard form: 9.3 x 107

Changing it Back We also need to be able to change numbers from standard form back to the original number. To do this we reverse the process. Example: write the number 4.2 x 104 as an ordinary number. In this case the value for n = 4. So we need to move the decimal point 4 places.

Changing it Back 4.2 x 104 can be written as 4.2000 x 104 which becomes 42 000 Remember 104 means 10 000, so we are really multiplying 4.2 by 10 000.

What about Small Numbers? We can also use standard form to represent small numbers. For example: write the number 0.000 058 in standard form. We use the same method as before. Begin by moving the decimal point to get a number between 1 and 10 0.000 058 becomes 5.8 This gives the value for a

Small Numbers Next we count the number of places that we moved the decimal point. In this case 5. But we have had to move the decimal point in the opposite direction so the value for n is -5 We can now write the number in standard form: 5.8 x 10-5

Changing it Back We also need to be able to change small numbers from standard form back to the original number. To do this we reverse the process. Example: write the number 8.1 x 10-5 as an ordinary number.

Changing it Back In this case the value for n = -5. So we need to move the decimal point 5 places. Remember negative indices make a small number 8.1 x 10-5 becomes 0.000 081