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This is a mathematical statement of the claim made above Square both sides of the equation in the previous line Use arithmetic to manipulate the equation in the previous line Multiply each side of the equation in the previous line by q2 Divide each side of the equation in the previous line by 2 Integer-squared rule Integer-squared rule We know this because q2 = p2/2, and q2 is an integer This follows from the definition of even integer According to the odd-integer-squared rule, if p were odd, then p2 would be odd; but p2 is even This follows from the definition of even integer This will make things a little simpler Multiply each side of the equation in the previous line by 2 Substitute 2t for p in the equation 2q2 = p2 from earlier in this proof Simplify the right-hand side of the equation in the previous line Divide each side of the equation in the previous line by 4 Integer-squared rule This follows from the previous two lines This follows from the definition of even integer According to the odd-integer-squared rule, if q were odd, then q2 would be odd; but q2 is even This follows from the definition of even integer Part of the assumption we made at the beginning of this proof We have proven both of these facts This follows from the statement immediately above this line (p /2)/(q /2) is in lower terms than p /q, but the two expressions represent the same quantity The preceding line contradicts our original assumption about p /q Reductio ad absurdum Mission accomplished

a measurement of the Doppler shift (in Hz) at the receiver indicated time By eqn (825) the Doppler measurement model is

Most of the properties of the natural numbers, integers, and rationals also apply to the reals To get ready for the algebra to come, let s put together a collection of these properties All the old rules are here, along with a few new or expanded ones Don t try to memorize these rules Just look them over for a little while You can always come back to this section for reference if you get stuck later on

s T Dr ( r )

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Naming variables You ll notice that some of the variables here have different letter names than we used before In algebra, real variables are usually given letters from near the end of the alphabet, especially t and after Other letters more often represent natural-number variables (n and m are especially common), integer variables, and rational-number variables But there s no absolute law about this Additive identity For every real number x,

Transistor Q1 controls the audible alarm by clamping the negative voltage returning through the ground path When a magnetic event occurs, the speaker emits an audible alarm, and the meter gives a visual indication of a changing magnetic field The adjustment of ALARM ADJUST potentiometer R24 can remove distortion from the sound of the speaker The author s prototype control/display unit is powered by eight AA cells: four cells provide positive voltage and four cells provide negative voltage As an alternative, the monitor can be powered by rechargeable nickel-cadmium cells

where t arguments have been dropped to simplify notation, the temporal i i di erences for E i , Ir , and Tr have been dropped because they are small relative to the other terms, ( r ) represents the measurement error due to multipath and receiver noise, and ti = ti ( r ) ti ( r T ) is the (uncorrected) satellite clock drift rate The symbol ti is used instead i as a reminder to the reader to correct the Doppler measurement for of t the satellite clock drift rate as described in eqn (835) Assuming that the line-of-sight vector from the satellite to the user s h = pr ps is available from the position solution, a linearized model for

Multiplicative identity For every real number x,

Additive inverse For every real number x,

(834)

In order to begin constructing the Aurora Monitor, you will need to secure a clean will lit work table or work bench so you can spread out all the tools, project components, charts and diagrams You will need to locate a small 27 to 33 watt pencil-tipped soldering iron, a small roll of 60/40 rosin core solder, a small jar of Tip Tinner, soldering iron tip cleaner/dresser, from your local Radio Shack store A few small hand tools

Multiplicative inverse (reciprocal) For every nonzero real number x,

The satellite velocity computation is described in Section C4 The satellite clock drift rate c ti is predicted by the broadcast model to be af 1 T , see Section C1 The Doppler residual measurement is de ned as

Commutative law for addition For all real numbers x and y,

s s Dr = Dr + h vs + caf 1 T,

Commutative law for multiplication For all real numbers x and y,

+6 V R24 Out In S2 R6 R8 R12 R14 R21 R13 R15 J2 +6 V 4 U2 6 5 2 3 3 8 R9 2 U3 4 +6 V C10 6 V R11 6 V C4 C5 C6 + B1 S3 B2 + S1 R17 S4 R18 D4 6 7 6 2 U4:A + 4 3 U4:B + 10 + C8 R19 R7 R10 C9 +6 V 9 1 13 R16 12 c D3 R23 Q3 b c 6 V b Q2 e C7 R20 c c R22 T1 b +6 V Q1 c 6 V C11 Recorder J3 C12 M1

Associative law for addition For all real numbers x, y, and z,

(835)

(x + y) + z = x + (y + z)

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