[et_pb_section fb_built=”1″ _builder_version=”4.9.10″ _module_preset=”default” custom_padding=”||0px|||” global_module=”299″ saved_tabs=”all”][et_pb_row _builder_version=”4.9.10″ _module_preset=”default”][et_pb_column type=”4_4″ _builder_version=”4.9.10″ _module_preset=”default”][et_pb_text _builder_version=”4.9.10″ _module_preset=”default” text_font=”Arial||||||||” text_font_size=”17px” min_height=”22.2px” custom_margin=”32px||-7px|||” custom_padding=”||0px|||” text_text_shadow_style=”preset2″ text_text_shadow_blur_strength=”0.41em”]<\/p>\n
3<\/span><\/b>.\u00a0 Mathematics<\/strong><\/span><\/span><\/p>\n [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row column_structure=”1_3,2_3″ _builder_version=”4.9.10″ _module_preset=”default” custom_padding=”||0px|||”][et_pb_column type=”1_3″ _builder_version=”4.9.10″ _module_preset=”default”][et_pb_text _builder_version=”4.9.10″ _module_preset=”default”]<\/p>\n 3.1\u00a0 \u00a0 \u00a0Mathematics<\/p>\n [\/et_pb_text][et_pb_image src=”https:\/\/vk6gmd.com.au\/wp-content\/uploads\/2021\/07\/graph1.png” title_text=”graph1″ _builder_version=”4.9.10″ _module_preset=”default”][\/et_pb_image][et_pb_image src=”https:\/\/vk6gmd.com.au\/wp-content\/uploads\/2021\/07\/graph2.png” title_text=”graph2″ _builder_version=”4.9.10″ _module_preset=”default”][\/et_pb_image][\/et_pb_column][et_pb_column type=”2_3″ _builder_version=”4.9.10″ _module_preset=”default”][et_pb_text _builder_version=”4.9.10″ _module_preset=”default” min_height=”34.6px” custom_padding=”||2px|||”]<\/p>\n [\/et_pb_text][et_pb_toggle title=”Addition and Subtraction” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n In any maths problem it must be remembered that a number without a plus or minus sign written against it is assumed to be a positive number.\u00a0<\/p>\n 5 + 5 = 10\u00a0 is really (+5) + (+5) = (+10)<\/p>\n Every number has a decimal point.\u00a0 A whole number such as 6 is six point zero (6\u22190), and for convenience sake we normally don\u2019t show the decimal zero.\u00a0 However it is just as correct to write 6\u22190000000000 as 6, the number is the same.<\/a><\/p>\n When adding<\/em> numbers of the same sign<\/em> the result will have the same sign as the numerals.\u00a0<\/p>\n 5+2 = 7\u00a0 or (+5) + (+2) = (+7)<\/p>\n (-5) + (-2)\u00a0 =\u00a0 -7<\/p>\n When subtracting<\/em> numbers of the same sign <\/em>the result will not always be as obvious. We are all familiar with\u00a0 five minus three equals two\u00a0 or (+5) \u2013 (+3) = (+2).<\/p>\n (-5) \u2013 (-2) = (-3) may not be a major problem but<\/p>\n (-2) \u2013 (-5) = (+3)\u00a0 could be confusing.<\/p>\n It may be easier to treat all subtraction problems as additions after working the signs.<\/em><\/p>\n Like signs may be treated as a plus.\u00a0 That is to say \u2013 (-5) is the same as +5.\u00a0<\/p>\n Using the above example :<\/p>\n (-2) \u2013 (-5) the \u2013 (-5) being like signs become a + and we have<\/p>\n (-2)\u00a0 + 5 and the answer becomes a little more obvious as +3<\/p>\n Similarly (-5) \u2013 (-3) becomes (-5) + 3 = -2<\/p>\n When adding values with different signs<\/em> remember:<\/p>\n Examples:<\/strong><\/p>\n Subtract the smaller number from the larger \u00a0[5-3 = 2]<\/p>\n Apply the sign of the larger number\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [-2]<\/p>\n Subtract the smaller number from larger \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [5-3 = 2]<\/p>\n Apply the sign of the larger number\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [+2 or 2]<\/p>\n [\/et_pb_toggle][et_pb_toggle title=”Fractions” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n <\/a>The terms numerator and denominator are used to indicate a part of a fraction. The numerator is the part of the fraction above the fraction bar.<\/span><\/p>\n The denominator is the part of the fraction below the fraction bar.<\/p>\n 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Numerator<\/p>\n \/ \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Fraction Bar<\/p>\n 2\u00a0\u00a0\u00a0\u00a0\u00a0 Denominator<\/p>\n Proper Fractions are those which indicate a part of a whole.<\/p>\n 1<\/sup>\/3<\/sub> indicates that the whole number is divided into 3 parts and only<\/p>\n one part is present.<\/p>\n It means the same as 1 divided by 3 and\u00a0 could be written as\u00a0 1\u00f73<\/p>\n <\/p>\n Often fractions can be simplified by dividing the numerator and denominator by the same number.<\/p>\n For example:<\/p>\n 3<\/sup>\/6<\/sub> is the same as 1<\/sup>\/2<\/sub>\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 both denominator and numerator have been divided by 3.<\/p>\n 8<\/sup>\/32<\/sub>\u00a0 =\u00a0 4<\/sup>\/16 <\/sub>\u00a0=\u00a0 2<\/sup>\/8<\/sub>\u00a0 =\u00a0 1<\/sup>\/4<\/sub> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 both have been divided by two and two and two again.<\/p>\n <\/a>Improper Fractions<\/p>\n Improper fractions are those that indicate a complete number and parts.<\/p>\n 4<\/sup>\/3 <\/sub>indicates that there is one whole (3<\/sup>\/3<\/sub>) and 1<\/sup>\/3<\/sub>.\u00a0 This can be simplified to 11<\/sup>\/3<\/sub><\/p>\n 40<\/sup>\/32\u00a0\u00a0 <\/sub>=\u00a0\u00a0 20<\/sup>\/16\u00a0 <\/sub>\u00a0=\u00a0 10<\/sup>\/8\u00a0 <\/sub>\u00a0=\u00a0\u00a0 5<\/sup>\/4\u00a0 <\/sub>\u00a0=\u00a0 11<\/sup>\/4 <\/sub>\u00a0\u00a0or<\/p>\n 40<\/sup>\/32\u00a0 <\/sub>\u00a0=\u00a0 1 8<\/sup>\/32\u00a0 <\/sub>= 14<\/sup>\/16\u00a0 <\/sub>=\u00a0 12<\/sup>\/8 <\/sub>\u00a0=\u00a0 11<\/sup>\/4<\/sub><\/p>\n <\/a>Decimal Fractions<\/p>\n A decimal is merely another way of expressing a fraction to the base ten.<\/p>\n 0\u22191 is 1 divided by ten, 1<\/sup>\/10 <\/sub>\u00a0or one tenth.<\/p>\n 0\u22195 is 5 divided by ten, 5<\/sup>\/10 <\/sub>or simplified to 1<\/sup>\/2<\/sub>\u00a0 or one half.<\/p>\n <\/p>\n The number of decimal places indicates the number of zeros in the denominator.<\/p>\n 0\u22191 is 1<\/sup>\/10 \u00a0\u00a0\u00a0<\/sub>:\u00a0\u00a0 0\u221901 is 1\/100\u00a0 <\/sub>\u00a0\u00a0:\u00a0\u00a0 0\u2219001 is 1<\/sup>\/1000 <\/sub>\u00a0\u00a0\u00a0:\u00a0\u00a0\u00a0 0\u221900001 is 1<\/sup>\/100000<\/sub>\u00a0 <\/sub><\/p>\n <\/p>\n As with whole numbers, 0\u22195 could be written as 0\u221950 or 0\u2219500,<\/p>\n Examples:<\/strong><\/p>\n 0\u221950 is 50<\/sup>\/100 <\/sub>which is the same as 5<\/sup>\/10<\/sub> or 0\u22195<\/p>\n 0\u2219500 is 500<\/sup>\/1000 <\/sub>which is the same as 50<\/sup>\/100 <\/sub>\u00a0(or 0\u221950) or\u00a0 5<\/sup>\/10<\/sub> (or 0\u22195).<\/p>\n [\/et_pb_toggle][et_pb_toggle title=”Powers of Ten” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n As we saw when squaring a number ten squared is one hundred (10<\/span>2 <\/sup>\u00a0= 100).<\/span><\/p>\n <\/a>This can be expanded by using higher exponents such as 103<\/sup> equals 1000<\/a><\/p>\n (10*10*10.\u00a0\u00a0\u00a0 10*10 = 100; 100*10 = 1000).<\/p>\n 106<\/sup> = 1000000 ( [1] 10*10 = 100;\u00a0 [2] 100*10 = 1000;\u00a0 [3] 1000*10 = 10000;\u00a0<\/p>\n [4] 10000*10 = 100000;\u00a0 [5]100000*10 = 1000000.<\/p>\n Note:<\/strong> \u00a0The exponent does not<\/em> indicate the number of times the number is multiplied but the number of times the number is used.<\/p>\n In 103<\/sup> there are only two \u2018*\u2019 steps but the 10 is used three times and in the 106<\/sup> there are five \u2018*\u2019 steps. the first step uses 10 twice (10*10) and each \u2018*\u2019 step after that uses 10 once so only five \u2018*\u2019 steps are used to multiply 10 by itself six times.\u00a0<\/p>\n These exponents with a base 10 may be used with another number, such as 6×103<\/sup>.\u00a0 This is 6*10*10*10 or 6000.\u00a0<\/p>\n (6*10 = 60; 60*10 = 600; 600*10 = 6000.\u00a0 Again the exponent shows how many times 10 is used in the multiplication process.\u00a0<\/p>\n Exponents may also be used in the negative, such as 10-3<\/sup>.\u00a0 The minus sign in front of the exponent indicates that the number is divided by 10 to the power shown.\u00a0 In this case,<\/p>\n divided by 103<\/sup> or 1000, so 10-3<\/sup> is 1<\/sup>\/1000<\/sub>.\u00a0 If combined with another number (5*10-3<\/sup>) it indicates the number (5) divided 1000 or 5<\/sup>\/1000<\/sub>.<\/p>\n Another way of writing exponents with a base 10<\/em> is to use the letter E in place of the 10.<\/p>\n 5*103<\/sup> could be written as 5E3 meaning five multiplied by ten to the power of three.<\/p>\n Note the E means a power of ten<\/em> so 2E2 is NOT 22<\/sup> (4) but is 2*102<\/sup> (200).\u00a0<\/p>\n Convert 6 000 000 000\u00a0\u00a0\u00a0 (There are nine zeros = E9) = 6E9<\/p>\n Convert 3E6\u00a0 (E6 indicates six zeros)\u00a0 = 3 000 000<\/p>\n NOTE:\u00a0 In electronics we usually work with exponents in the multiple of 3 (E3, E6,E9 etc) because they equate to the SI Prefixes (to be explained shortly). but remember that exponents may be 2E2 (200) which is the same as 0\u22192E3 (200) or 20E1 (200).<\/p>\n [\/et_pb_toggle][et_pb_toggle title=”S.I. Units” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n <\/a>When using the decimal system any unit (meter, litre, gram etc) can be prefixed with an indicator to show larger or smaller amounts. Many of these are in common usage such as kilometre (1000m), millilitre (1<\/sup>\/1000<\/sub> ).\u00a0 The prefixes remain the same, regardless of the unit or what is being measured.\u00a0 Kilo can be applied to any measurement meaning a thousand of that unit whether it is a gram, meter, litre or snazzle. The only place where these are not used is in time measurement of greater than a second and less than a year.<\/p>\n They can however be used to indicate time of less than a second.\u00a0\u00a0<\/p>\n The prefixes go as high as Exa (1E18) and as small as atto (1E-18).\u00a0 In this course we are only concerned with units between Tera\u00a0 (1E12 and pico (1E-12).<\/p>\n Prefix\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Number\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Power of ten<\/p>\n Tera (T)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 1 000 000 000 000\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a01E12\u00a0\u00a0 (1x*10^12 or 1*1012<\/sup>)<\/p>\n Giga (G)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 1 000 000 000\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1E9\u00a0\u00a0\u00a0\u00a0 (1*10^9\u00a0 or\u00a0 1*109<\/sup>)<\/p>\n Mega (M)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1 000 000\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1E6\u00a0\u00a0\u00a0\u00a0 (1*10^6\u00a0 or\u00a0\u00a0 1*106<\/sup>)<\/p>\n Kilo\u00a0 (k)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1 000\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1E3\u00a0 \u00a0\u00a0\u00a0(1*10^3\u00a0 or\u00a0\u00a0\u00a0 1*103<\/sup>)<\/p>\n Unit\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 1<\/p>\n Milli (m)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<\/sup>\/1000 <\/sub>or 0\u2219001\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a01E-3\u00a0\u00a0\u00a0 (1*10^-3\u00a0 or\u00a0 1*10-3<\/sup>)<\/p>\n Micro (\u03bc or u) \u00a0\u00a0\u00a0\u00a0\u00a0 1<\/sup>\/1 000 000 <\/sub>or 0\u2219000 001\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a01E-6\u00a0\u00a0\u00a0 (1*10^-6\u00a0 or\u00a0 1*10-6<\/sup>)<\/p>\n Nano (n)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 1<\/sup>\/1 000 000 000<\/sub> or 0.000 000 001\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1E-9\u00a0\u00a0\u00a0 (1*10^-9\u00a0 or\u00a0\u00a0 1*10-9<\/sup>)<\/p>\n Pico\u00a0 (p)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<\/sup>\/1 000 000 000 000 <\/sub>or\u00a0 0.000 000 000 001\u00a0 \u00a0 \u00a0 1E-12\u00a0 (1*10^-12 or\u00a0 1*10-12<\/sup>)<\/p>\n When using these prefixes it is usually easier to work in decimals than fractions.<\/p>\n <\/a>Converting S.I. Units<\/strong><\/p>\n To convert one unit to another it is a matter of moving the decimal place to the left or right.\u00a0 To convert 5000m to km the decimal place moves three places to the left leaving us with 5km.\u00a0 Starting with 5 0 0 0 \u2219 0m\u00a0<\/p>\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 5 0 0\u22190\u00a0\u00a0\u00a0 is one place<\/p>\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 5 0\u22190\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 is two places<\/p>\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 5\u22190\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 is three places<\/p>\n If converting in the other direction, 5km to m the decimal is moved in the opposite direction so 5\u22190km becomes 5000\u22190m.\u00a0 The decimal point and the following zero are normally not written in so 5km = 5000m.\u00a0<\/p>\n Once the units scale is known the rule becomes:<\/p>\n If moving upward on the scale<\/strong> the unit (1) the decimal point moves to the left three places<\/strong> for each prefix.<\/p>\n So if converting 7pF to nF.\u00a0 The scale goes pico, nano, micro, milli, 1, kilo etc so we are moving upward on the scale by one prefix so\u00a0 the decimal point moves to the left three places.\u00a0<\/p>\n 7\u22190; move one place = 0\u22197 ; the second place\u00a0 =\u00a0 0\u221907 ; third place = 0\u2219007nF<\/p>\n Convert 600kV to MV.\u00a0 Moving upward on the scale by one prefix the decimal point moves to the left by three places.\u00a0 600; moving one place = 60\u22190;\u00a0 second place = 6\u221900; third place = 0\u2219600.\u00a0 600kV = 0\u22196MV<\/p>\n Convert 3nF to mF.\u00a0 In this case we are moving through 2 prefixs (n, \u03bc, m) so the decimal moves six places to the left.\u00a0 3nF = 0\u2219000 003 mF.\u00a0<\/p>\n If moving downward on the scale<\/strong> the decimal moves to the right <\/strong>three places for each prefix.<\/p>\n Convert 4MV to V.\u00a0 Moving down the scale move by two prefixes move the decimal point six places to the right.\u00a0 One place = 40; two places = 400; three places = 4 000; four places = 40 000; five places = 400 000; six places = 4 000 000.\u00a0 4MV = 4 000 000V<\/p>\n Convert 500\u03bcH to pH.\u00a0 Moving down on the scale by two prefixs \u2013 move the decimal 6 places to the right.\u00a0 One place = 5 000; two places = 50 000;\u00a0 three = 500 000; four = 5 000 000;\u00a0 five = 50 000 000 and six = 500 000 000.\u00a0 500\u03bcH =\u00a0 500 000 000pH<\/p>\n Convert 6km to m.\u00a0 Moving down on the scale the decimal point moves to the right.\u00a0 One prefix so it moves three places.\u00a0 6km = 6 000m<\/p>\n [\/et_pb_toggle][et_pb_toggle title=”Square of a Number” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n <\/a>A square of a number is the number multiplied by itself and is indicated by a superscript figure 2 following the number (called the exponent). For example 3 squared would be written 3<\/span>2<\/sup>.\u00a0 Because putting in a superscript is not always possible it may be written as 3^2 (the ^ indicates following number(s) are in superscript).<\/span><\/p>\n This indicates that the 3 is multiplied by itself that is 3*3 = 9.<\/p>\n 102<\/sup> = 100 or 10*10<\/p>\n Note:\u00a0 The number is not<\/em> multiplied by the exponent but by itself.<\/p>\n <\/a>Square Root of a Number<\/p>\n The square root of a number is the opposite of the square of a number.\u00a0 That is to say the square root (written as \u221a ) of a number is the number which is multiplied by itself to give the original number.<\/p>\n For example \u221a4 is 2 (because 2*2=4)\u00a0 or the \u221a9 = 3 (3*3=9).<\/p>\n For small simple numbers these can be calculated from \u2018times tables\u2019 learnt in primary schools but for larger or more complicated numbers a calculator is required.<\/p>\n [\/et_pb_toggle][et_pb_toggle title=”Operations with exponents” _builder_version=”4.9.10″ _module_preset=”default” title_font=”|700|||||||” title_font_size=”14px” custom_margin=”-35px|||||”]<\/p>\n Multiplication with Exponents<\/strong><\/a><\/p>\n When multiplying two numbers with exponents, the numbers are multiplied and the exponents are added.<\/p>\n 2E3*2E3\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 The numbers (2*2) are multiplied as normal (4)<\/p>\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The exponents (E3 and E3) are added (E6)<\/p>\n giving a result of\u00a0 4E6.<\/p>\n This can be proved by converting the original problem to whole numbers and multiplying:<\/p>\n 2E3*2E3 = 2000*2000 = 4 000 000\u00a0 or\u00a0 4E6<\/p>\n Examples:<\/strong><\/p>\n Resolve\u00a0 5E3*2E6\u00a0<\/p>\n Multiply the numbers (5*2) = 10 and add the exponents (E3+E6) = E9 gives 10E9<\/p>\n Resolve 3E6*2E-3<\/p>\n Mulitply the numbers (3*2) = 6 and add the exponents (E6+E-3) = E3 gives 6E3<\/p>\n Resolve 4E-6*2E3<\/p>\n Multiply the numbers (4*2) = 8 and add the exponents E-6+E3 = E-3 gives 8E-3<\/p>\n\n
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<\/a>Proper Fractions<\/span><\/h2>\n